Morita equivalence of semirings with local units
In this paper we study some necessary and sufficient conditions for two semirings with local units to be Morita equivalent. Then we obtain some properties which remain invariant under Morita equivalence.
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| Цитувати: | Morita equivalence of semirings with local units / M. Das, S. Gupta, S.K. Sardar // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 1. — С. 37–60. — Бібліогр.: 17 назв. — англ. |
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Das, M. Gupta, S. Sardar, S.K. 2023-03-11T13:13:46Z 2023-03-11T13:13:46Z 2021 Morita equivalence of semirings with local units / M. Das, S. Gupta, S.K. Sardar // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 1. — С. 37–60. — Бібліогр.: 17 назв. — англ. 1726-3255 DOI:10.12958/adm1288 2020 MSC: 16Y60, 16Y99 https://nasplib.isofts.kiev.ua/handle/123456789/188676 In this paper we study some necessary and sufficient conditions for two semirings with local units to be Morita equivalent. Then we obtain some properties which remain invariant under Morita equivalence. The first author is grateful to CSIR, Govt. of India, for providing research support. The authors are very much thankful to the learned referee for his meticulous referring and subsequent detection of some errors both in mathematics as well as presentation whose compliance has made the paper what it is now. The authors are also grateful to Prof. L. Márki and Prof. P. N. Ánh of Alfréd Rényi Institute of Mathematics, Hungary for a necessary discussion in order to resolve one issue raised by the learned referee. Lastly the authors would like to acknowledge the encouragement provided by Prof. Y. Katsov of Hanover College, USA at the time of preparation of the paper. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Morita equivalence of semirings with local units Article published earlier |
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Morita equivalence of semirings with local units |
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Morita equivalence of semirings with local units Das, M. Gupta, S. Sardar, S.K. |
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Morita equivalence of semirings with local units |
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Morita equivalence of semirings with local units |
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Morita equivalence of semirings with local units |
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Morita equivalence of semirings with local units |
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morita equivalence of semirings with local units |
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Das, M. Gupta, S. Sardar, S.K. |
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Das, M. Gupta, S. Sardar, S.K. |
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2021 |
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Algebra and Discrete Mathematics |
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In this paper we study some necessary and sufficient conditions for two semirings with local units to be Morita equivalent. Then we obtain some properties which remain invariant under Morita equivalence.
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1726-3255 |
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https://nasplib.isofts.kiev.ua/handle/123456789/188676 |
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Morita equivalence of semirings with local units / M. Das, S. Gupta, S.K. Sardar // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 1. — С. 37–60. — Бібліогр.: 17 назв. — англ. |
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2025-11-25T20:57:21Z |
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2025-11-25T20:57:21Z |
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1850543881309913088 |
| fulltext |
“adm-n1” — 2021/4/10 — 20:38 — page 37 — #41
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 31 (2021). Number 1, pp. 37–60
DOI:10.12958/adm1288
Morita equivalence of semirings with local units
M. Das, S. Gupta, and S. K. Sardar
Communicated by A. V. Zhuchok
Abstract. In this paper we study some necessary and
sufficient conditions for two semirings with local units to be Morita
equivalent. Then we obtain some properties which remain invariant
under Morita equivalence.
1. Introduction
The classical Morita theory for rings has been recognized as one of
the most important and fundamental tools in studying the structure of
rings. In 1958 Morita [13] established the Morita equivalence theory for
rings with identity. In 1983, Abrams [1] made a first step in extending
the theory of Morita equivalence to rings without identity. He considered
rings with a commuting set of idempotents such that every element of the
ring admits one of these idempotents as a two-sided identity and studied
the equivalence of the categories of all unitary left modules of these rings.
Ánh and Márki [4] further generalized Abrams’ result to rings with local
units by weakening the condition of commutativity of idempotents. In the
year 2011, Katsov and Nam [10] transferred the ring theoretic approach
of Morita equivalence to semirings with identity. In [15], Sardar et al.
connected Morita equivalence of semirings with a new and equivalent
version of Morita context for semirings. Later Katsov et al. [11], Sardar
and Gupta [16, 17] independently studied some properties of semirings
The first author is grateful to CSIR, Govt. of India, for providing research support.
2020 MSC: 16Y60, 16Y99.
Key words and phrases: Morita equivalence, Morita context, Morita invariant,
semiring, semimodule.
https://doi.org/10.12958/adm1288
“adm-n1” — 2021/4/10 — 20:38 — page 38 — #42
38 Morita equivalence of semirings
which remain invariant under Morita equivalence. The aim of this paper
is to extend the theory to cover a wider range of semirings namely the
semirings with local units in the sense that any two elements of the semiring
have a common two sided identity. In order to develop this theory we
consider the category R-Sem consisting of all unitary left R-semimodules
M i.e., semimodules RM such that RM =M , where R is a semiring with
local units and say two such semirings R and S to be Morita equivalent
if the categories R-Sem and S-Sem are equivalent. Since for a semiring
A with identity, A-Sem coincides with the category A-SEM of all left
A-semimodules, our notion of Morita equivalence coincides with that of
semiring with identity [10]. Consequently, some of the results of Katsov et
al. [10] are encompassed in their counterparts obtained here. We organize
the paper as follows. In Section 2 we recall some necessary preliminaries
on semirings and semimodules. In Section 3 we define locally projective
unitaryR-semimodule and present some characterizing properties of locally
projective generators in semimodule categories. In Section 4 we develop
some tools to investigate some necessary and sufficient conditions for
R-Sem and S-Sem to be equivalent. Analogously to the case of semirings
with identity we show that two semirings with local units R and S are
Morita equivalent if and only if there exists a unitary Morita context
(R,S, P,Q, τ, µ) with τ, µ surjective. We also identify the semirings with
local units which are Morita equivalent to semirings with identity (cf.
Prop. 4.14). Finally we conclude the paper by studying some properties
of semirings preserved under Morita equivalence in Section 5.
2. Preliminaries
A semiring1 [6] is a nonempty set R on which operations of addition
and multiplication have been defined such that (1) (R,+) is a commutative
monoid with identity element 0, (2) (R, ·) is a semigroup, (3) multiplication
distributes over addition from either side, (4) 0r = 0 = r0 for all r ∈ R. A
left R-semimodule over a semiring R is a commutative monoid (M,+, 0M )
together with a scalar multiplication from R × M to M , denoted by
(r,m) 7→ rm, which satisfies the following identities: (1) (r + r′)m =
rm+ r′m, (2) r(m+m′) = rm+ rm′, (3) (rr′)m = r(r′m), (4) r0M =
0M = 0m for all r, r′ ∈ R and m,m′ ∈ M . Right R-semimodules and
R-S-bisemimodules are defined analogously. We will distinguish left and
right R-semimodules by writing RM and MR, respectively. Let M and N
1Although Golan called it a hemiring, we call it semiring in the present article.
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M. Das, S. Gupta, S. K. Sardar 39
be two left R-semimodules. Then a monoid homomorphism f :M → N is
called a left R-homomorphism if f(rm) = rf(m) for all r ∈ R and m ∈M .
The set of all R-morphisms from M to N is denoted by HomR(M,N), in
particular EndR(M) denotes the set of all R-morphisms from M to itself.
Right R-homomorphisms and bisemimodule homomorphisms are defined
analogously.
A semiring R (semimodule P ) is called additively cancellative [7] if
a+ x = a+ y implies x = y for all a, x, y ∈ R (respectively a, x, y ∈ P )
and called additively idempotent [7] if a+ a = a for all a ∈ R (respectively
a ∈ P ). If for each element a of a semiring R (semimodule P ) there exists
an element b ∈ R (respectively b ∈ P ) such that a+ b+a = a the semiring
(respectively semimodule) is said to be additively regular [6]. A semiring
R (semimodule P ) is said to be zero-sum free [7] if a + b = 0 implies
a = b = 0 for all a, b ∈ R (respectively a, b ∈ P ). A nonempty subset I of
a semiring R is called an ideal [6] of R if i+ j ∈ I and ri, ir ∈ I for any
i, j ∈ I and r ∈ R. A semiring (semimodule) is said to be Noetherian [6]
if any ascending chain of ideals (respectively subsemimodules) terminates.
Now we recall some preliminaries related to k-ideals and h-ideals. An
ideal I of a semiring R is called a k-ideal [8] (also called subtractive
ideal in [6]) of R if for x ∈ I, y ∈ R, x + y ∈ I implies y ∈ I. A
subsemimodule N of a semimodule P is called a k-subsemimodule1 (called
subtractive subsemimodule in [6]) of P if for x ∈ N, y ∈ P, x+ y ∈ N
implies y ∈ N . An ideal I of a semiring R is called an h-ideal [8] of
R if for y1, y2 ∈ I, x, z ∈ R, x + y1 + z = y2 + z implies x ∈ I. A
subsemimodule N of a semimodule P is called an h-subsemimodule [14] of
P if for y1, y2 ∈ N, x, z ∈ P, x+ y1 + z = y2 + z implies x ∈ N . The k-
closure [8] of an ideal I (a subsemimodule N) is denoted by I (respectively
N) and is defined by I = {x ∈ R | x+ i ∈ I for some i ∈ I} (respectively
N = {x ∈ P | x+ p ∈ N for some p ∈ N}). The h-closure [8] of an ideal I
(a subsemimodule N) is denoted by Ĩ (respectively Ñ) and is defined by
Ĩ = {x ∈ R | x+ y1 + z = y2 + z for some y1, y2 ∈ I, z ∈ R} (respectively
Ñ = {x ∈ P | x+ p1 + z = p2 + z for some p1, p2 ∈ N, z ∈ P}).
An ideal I of a semiring R defines a congruence BI on R, called the
Bourne congruence [6], given by rBIr
′ if and only if there exist a, a′ ∈ I
satisfying r + a = r′ + a′. Similarly I defines another congruence II on R,
called the Iizuka congruence [6], given by rIIr′ if and only if there exist
a, a′ ∈ I and s ∈ R satisfying r+ a+ s = r′ + a′ + s. A congruence ρ on a
semiring R is called a ring congruence [5] if the factor semiring R/ρ is a
1In the present article subtractiveness is replaced by k.
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40 Morita equivalence of semirings
ring. A subsemimodule N of a semimodule P defines a congruence BN on
P , called the Bourne congruence [6], given by pBNp
′ if and only if there
exist a, a′ ∈ N satisfying p + a = p′ + a′. Similarly N defines another
congruence IN on P , called the Iizuka congruence [6], given by pINp′ if and
only if there exist a, a′ ∈ N and p′′ ∈ P satisfying p+a+p′′ = p′+a′+p′′.
For preliminaries on category theory we refer to [2], [9] and [12].
We adopt the following notions from Ánh and Márki [4].
Definition 2.1. Let R be a semiring and E(R) be a set of idempotents
of R. Then R is said to be a semiring with local units if every finite subset
of R is contained in a subsemiring of the form eRe where e ∈ E(R) or
equivalently if for any finite number of elements r1, r2, . . . , rn ∈ R, there
exists e ∈ E(R) such that eri = ri = rie for all i = 1, 2, . . . , n. In this case
E(R) is a set of local units (slu) of R.
Here we give some examples of semirings with local units.
Example 2.2. 1. Suppose L is a distributive lattice with the least element
0 but with no greatest element1. Consider L together with the addition
+ and multiplication · defined by a+ b = sup{a, b} and a · b = inf{a, b}
respectively, for a, b ∈ L. Then (L,+, ·) is a semiring with additive identity
0 but with no multiplicative identity. But it is a semiring with local units,
as for any two elements a, b ∈ L, by the absorption law, a · (a+ b) = a =
(a+ b) · a and b · (a+ b) = b = (a+ b) · b, i.e., a+ b acts as the common
two-sided identity of a and b.
2. Let S be a semiring with identity, X be an infinite set and R =
{f | f : X → S has finite support}. Then R together with the operations
(f + g)(x) := f(x) + g(x) and (fg)(x) := f(x)g(x) for f, g ∈ R and
x ∈ X is a semiring without multiplicative identity. But it is a semiring
with local units in view of the following reasons. Suppose f, g ∈ R with
finite supports supp(f) and supp(g) respectively, define h : X → S
by h(x) = 1 if x ∈ supp(f) ∪ supp(g) and h(x) = 0 otherwise, then for
x ∈ supp(f), fh(x) = f(x)h(x) = f(x) and for x ∈ X \supp(f), fh(x) =
f(x)h(x) = 0 · h(x) = 0 = f(x). By a similar argument hf = f and hence
fh = f = hf and similarly gh = g = hg, i.e., h acts as a two-sided identity
of f and g.
Definition 2.3. A left R-semimodule M over R is said to be unitary if
RM =M i.e., for each m ∈M , there exist r1, r2, . . . , rn ∈ R and m1,m2,
. . . ,mn ∈M such that m = r1m1 + r2m2 + · · ·+ rnmn.
1(N, lcm, gcd), (N,max,min), where N is the set of all non-negative integers, are
some examples of such lattices.
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M. Das, S. Gupta, S. K. Sardar 41
Remark 2.4. If R is a semiring with slu E and M is a unitary R-
semimodule then for each m ∈ M , m = r1m1 + r2m2 + · · · + rnmn for
some r1, r2, . . . , rn ∈ R, m1,m2, . . . ,mn ∈M . Now for r1, r2, . . . , rn ∈ R,
there exists e ∈ E such that eri = ri for all i = 1, 2, . . . , n, therefore
m =
∑n
i=1 rimi =
∑n
i=1 erimi = em. Thus for every finite subsetM ′ ⊂M
there exists an e ∈ E such that eM ′ =M ′.
By R-Sem we denote the category of unitary left R-semimodules
together with usual R-morphisms.
3. Locally projective generators
In what follows unless otherwise mentioned any semiring is with local
units and homomorphisms of semimodules are written opposite the scalars.
Definition 3.1. [6] Let R be a semiring with local units. A semimodule
P ∈ R-Sem is said to be projective if for a surjective R-morphism φ :M →
N and an R-morphism α : P → N in R-Sem there exists an R-morphism
α : P →M satisfying αφ = α.
Recall that [10], for any R-semimomodule P , the trace ideal tr(P ) :=∑
q∈HomR(P,R) Pq ⊆ R.
Proposition 3.2. Let R be a semiring with local units. For any semi-
module P ∈ R-Sem, the following are equivalent:
(1) tr(P ) = R.
(2) There exists a surjective R-morphism φ :
⊕
I P → R for some index
set I.
(3) For every semimodule M ∈ R-Sem, there exists a surjective R-
morphism ψ :
⊕
Λ P →M for some index set Λ.
Proof. (1) ⇒ (2) Consider the family of all R-morphisms, fα : P →
R. Now if we set I = HomR(P,R), then the coproduct induced map
f =
⊕
I fα :
⊕
I P → R is a surjective R-morphism since (
⊕
I P )f =∑
fα∈I
Pfα = tr(P ) = R.
(2) ⇒ (3) Suppose there exists a surjective R-morphism φ :
⊕
I P → R
for some index set I. Now let M ∈ R-Sem, then for each m ∈M consider
the map ρm : R → M defined by r 7→ rm, then the coproduct induced
map ρ =
⊕
m∈M ρm :
⊕
M R → M is a surjective R-morphism since
(
⊕
M R)ρ =
∑
m∈M Rρm =
∑
m∈M Rm = M . Then the direct sum
φ′ =
⊕
M φ :
⊕
M (
⊕
I P ) →
⊕
M R is a surjection. Hence ψ = φ′ρ :⊕
Λ P →M is a surjective R-morphism where Λ =
⋃̇
MI
1.
1
⋃̇
denotes the disjoint union
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42 Morita equivalence of semirings
(2) follows trivially from (3).
(2) ⇒ (1) Suppose there exists a surjective R-morphism φ :
⊕
I P → R
for some index set I. Consider the natural inclusions ιi : P →
⊕
I P for all
i ∈ I. Now for each i ∈ I, let φi = ιiφ, then R = (
⊕
I P )φ =
∑
i∈I Pφi ⊆∑
q∈HomR(P,R) Pq = tr(P ). Hence tr(P ) = R.
Definition 3.3. A semimodule P ∈ R-Sem is said to be a generator for
the category R-Sem if P satisfies the equivalent conditions of Prop. 3.2.
Let R be a semiring with local units. Let M be a unitary left R-
semimodule and A be a subset of M . Then RA = {r1a1 + r2a2 + · · · +
rnan | n ∈ N, ri ∈ R, ai ∈ A, for all i = 1, 2, . . . , n} is the subsemimodule
generated by A. If A generates all of the semimodule M then A is a set
of generators for M . A unitary R-semimodule M is said to be finitely
generated if it has a finite set of generators.
We skip the proof of the following proposition as it is analogous to
that of its counterpart in module theory (see [3, Proposition 10.1]).
Proposition 3.4. If M is a finitely generated unitary left R-semimodule
then the following hold:
(1) For every set A of subsemimodules of M that spans M , there is a
finite set F ⊆ A that spans M .
(2) Every semimodule that generates M finitely generates M .
Lemma 3.5. Retract of a projective unitary R-semimodule is projective.
Proof. Using the definition of projectivity in R-Sem (cf. Def. 3.1), the
result follows easily from the proof of [9, Prop. 1.7.30] by replacing the
notion of epimorphism by surjectivity.
The next result is simply a restatement of Prop 17.19 of [6] in the
special case of the category R-Sem of R-semimodules where R is a semiring
with local units.
Proposition 3.6. If {Pi| i ∈ Ω} is a family of unitary left R-semimodules
then P =
⊕
i∈Ω Pi is projective if and only if each Pi is projective.
Proposition 3.7. RP is a finitely generated projective unitary semimodule
if and only if there exists an idempotent e ∈ R such that P is a retract of
(Re)n, n > 1.
Proof. Suppose P is a finitely generated projective unitary semimodule.
If P = {0} then the zero map θ : Re → P is a retraction in R-Sem. So
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M. Das, S. Gupta, S. K. Sardar 43
we assume that P 6= {0} and {p1, p2, . . . , pn} is a spanning set of RP .
Then there exists e2 = e ∈ R such that epi = pi for all i = 1, 2, . . . , n.
Consider φ : (Re)n → P defined by (x1, x2, . . . , xn)φ =
∑n
i=1 xipi. Since
for any p ∈ P there exist r1, r2, . . . , rn ∈ R such that p =
∑n
i=1 ripi,
(r1e, r2e, . . . , rne)φ =
∑n
i=1 riepi =
∑n
i=1 ripi = p. Thus φ is onto. Now
P being projective there exists h : P → (Re)n such that hφ = idP .
Conversely, suppose ψ : (Re)n → P is a retraction in R-Sem. Let f : A→
B be a surjection in R-Sem and g : Re→ B be an R-morphism. Define
g : Re→ A by t 7→ ta, where t ∈ Re and a ∈ A such that af = eg (if there
are more than one a ∈ A with af = eg then we choose any one of them
and fix it throughout). Then gf = g, hence Re is projective. Therefore by
Prop. 3.6, (Re)n is projective and from Lemma 3.5, RP is projective. Also
since (Re)n has a finite spanning set {ei : i = 1, 2, . . . , n}, where each
ei = (0, . . . , e, . . . , 0), with e in the i−th place for all i = 1, 2, . . . , n, P is
spanned by {eiψ : i = 1, 2, . . . , n}. Thus RP is finitely generated.
The notions introduced in the following two definitions are adopted
from Ánh and Márki [4].
Definition 3.8. Let I be a partially ordered set such that for each i, j ∈ I
there exists k ∈ I with i, j 6 k and (Mi)i∈I a family of unitary R-
semimodules. Then (Mi)i∈I is said to be a direct system if for any i 6 j
we have R-morphism φij :Mi →Mj such that φii = 1Mi
for all i ∈ I and
φijφjk = φik for i 6 j 6 k.
Moreover a direct system (Mi)i∈I is called a split direct system if for
each i 6 j in I there exists ψji : Mj → Mi such that φijψji = 1Mi
and
ψkjψji = ψki for i 6 j 6 k. In this case it follows that ψii = 1Mi
.
Definition 3.9. A unitary R-semimodule M is said to be locally projective
if it is the direct limit of a split direct system (Mi)i∈I consisting of
subsemimodules that are finitely generated projective.
Proposition 3.10. The R-semimodule RR is a locally projective genera-
tor.
Proof. LetE be a set of local units ofR. Define a binary relation 6 on E by
e 6 f if and only if ef = fe = e. Then clearly 6 is a partial order relation
on E and R being a semiring with local units (E,6) is an upward directed
set. Now for each idempotent e ∈ R and for each pair e, f ∈ R with e 6 f
consider the map ψfe : Rf → Re given by r′ 7→ r′e, where r′ ∈ Rf and the
natural inclusion maps φe : Re→ R and φef : Re→ Rf . Then (Re)e∈E
is a split direct system in R-Sem and R = lim−→ERe where Re is finitely
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44 Morita equivalence of semirings
generated projective (as seen in the proof of Prop. 3.7) R-semimodule
for each e ∈ E. Hence R is locally projective. Also for any unitary R-
semimodule M and for each m ∈ M consider the map ρm : R → M
defined by r 7→ rm, then we have ρ =
⊕
m∈M ρm :
⊕
M R → M , where
(
⊕
M R)ρ =
∑
m∈M Rρm =
∑
m∈M Rm = M , which implies that ρ is a
surjection. Therefore R is a generator in R-Sem.
Proposition 3.11. Let M be a locally projective unitary R-semimodule,
then every finitely generated subsemimodule P of M is contained in a
finitely generated projective subsemimodule of M .
Proof. Let M be a locally projective unitary R-semimodule. Then there
exists a split direct system (cf. Definition 3.8) (Mi)i∈I of finitely generated
projective subsemimodules of M such that M = lim−→IMi. Let M ′ =
∪̇Mi/ρ, where ρ on ∪̇Mi is given by (x, i)ρ(y, j) if and only if there exists
k ∈ I, i, j 6 k such that xφik = yφjk, where i, j ∈ I, x ∈ Mi, y ∈ Mj .
Using the existence of ψj′i′ for each i′, j′ ∈ I, i′ 6 j′, it then easily follows
that (x, i)ρ(y, j) if and only if xφik = yφjk for all k ∈ I, i, j 6 k. Now
it is a routine matter to verify that M ′ together with the family of R-
morphisms φi : Mi → M ′ given by x 7→ [(x, i)]ρ is the direct limit of
the split direct system (Mi)i∈I . Let P be a subsemimodule of M with a
finite spanning set {p1, p2, . . . , pn}. Then identifying M with M ′ we have
pk = [(xk, ik)]ρ for each k = 1, 2, . . . , n where ik ∈ I, xk ∈Mik . Let t ∈ I
such that ik 6 t for all k = 1, 2, . . . , n. Then for each k = 1, 2, . . . , n we
have pk = xkφik = xkφiktφt ∈ Mtφt. Therefore P ⊆ Mtφt ∼= Mt, where
Mt is a finitely generated projective subsemimodule of M . Hence the proof
is complete.
We observe that if R and S are semirings with local units and US
and RVS are unitary then HomS(U, V ) is a left R-semimodule by putting,
for φ ∈ HomS(U, V ) and r ∈ R, (rφ)(u) = rφ(u) for u ∈ U . The sub-
semimodule RHomS(U, V ) is the largest unitary R-subsemimodule of
HomS(U, V ).
Proposition 3.12. Suppose R is a semiring with slu E. Then
ρ : 1R-Sem → RHomR(R,_) is a natural isomorphism where for each
M ∈ R-Sem, ρM :M → RHomR(R,M) is given by m 7→ mρM (r 7→ rm).
For M ′ ∈ R-Sem and f ∈ HomR(M,M ′), ρf : RHomR(R,M) →
RHomR(R,M
′) is given by γ 7→ γf .
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M. Das, S. Gupta, S. K. Sardar 45
Proof. Clearly ρM is an R-morphism. Also the following diagram com-
mutes:
M
ρM //
f
��
RHomR(R,M)
ρf
��
M ′
ρM′
// RHomR(R,M
′)
since r((mρM )ρf ) = r(mρMf) = (rm)f = r(mf) = r((mf)ρM ′). Hence
ρ is a natural transformation. For M ∈ R-Sem, let m1, m2 ∈ M , such
that m1ρM = m2ρM . Now since there exists e ∈ E such that em1 =
m1, em2 = m2, we have m1 = e(m1ρM ) = e(m2ρM ) = m2. Hence ρM is
injective. Now let rf ∈ RHomR(R,M) and suppose (r)f = m ∈M then
for any t ∈ R, t(mρM ) = tm = t((r)f) = (tr)f = t(rf) i.e., mρM = rf .
Thus ρ is a natural isomorphism.
Definition 3.13. [10] Let MR be a right R-semimodule and RN be a left
R-semimodule. If F is the free N0-semimodule generated by the cartesian
productM×N and σ is the congruence on F generated by all ordered pairs
having the form ((m+m′, n), (m,n)+(m′, n)), ((m,n+n′), (m,n)+(m,n′))
and ((mr, n), (m, rn)) with m,m′ ∈MR, n, n′ ∈R N and r ∈ R, then the
factor semimodule F/σ is defined to be the tensor product of M and N
and is denoted by M ⊗RN . When there is no confusion over the semiring,
we denote the tensor product as M ⊗N and the class containing (m,n)
by m⊗ n.
Proposition 3.14. Suppose R is a semiring with slu E and M ∈ R-Sem.
Then R⊗M ∼=M .
Proof. Suppose R is a semiring with slu E and M is a unitary R-
semimodule. Consider the map µ :M → R⊗M defined by m 7→ e⊗m,
where m ∈ M and e ∈ E such that em = m. First we show that the
definition is independent of the choice of the idempotent e. Suppose e and
f are two idempotents in R such that em = m = fm. Let g ∈ E be a
common identity of e and f , then e ⊗m = ge ⊗m = g ⊗ em = g ⊗m.
Similarly f ⊗m = g⊗m, hence e⊗m = f ⊗m. Now it is a routine matter
to verify that µ is an R-morphism. Also consider the map ψ : R⊗M →M
defined by r ⊗m 7→ rm, where r ∈ R and m ∈ M . Clearly ψ is a well
defined R-morphism. Now for r ∈ R, m ∈M , we have
(r ⊗m)ψµ = (rm)µ = e⊗ rm = er ⊗m = r ⊗m,
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46 Morita equivalence of semirings
where e ∈ E such that er = r, i.e., erm = rm. Also
mµψ = (g ⊗m)ψ = gm = m,
where g ∈ E such that gm = m.
Hence µ is an isomorphism, i.e., R⊗M ∼=M .
Suppose R is a semiring with slu E(R) and RP is a unitary semimodule.
Let T be a subsemiring of EndR P having local units E(T ) such that
T EndR P = T and P ∈ Sem-T . Now consider the T − R bisemimodule
Q = T HomR(P,R)R. Then define:
τ : P ⊗Q→ R
p⊗ q 7→ pq
and µ : Q⊗ P → T
q ⊗ p 7→ qp (p′ 7→ (p′q)p)
It is routine to verify that the maps τ, µ are respectively R−R and T −T
bisemimodule morphisms. Also, there is a QPQ-associativity, i.e., for any
q, q′ ∈ Q and p′ ∈ P, q(p′q′) = (qp′)q′ since for any p ∈ P, p(q(p′q′)) =
(pq)(p′q′) = ((pq)p′)q′ = (p(qp′))q′ = p((qp′)q′) i.e., q(pq′) = (qp)q′.
In the notations introduced above, we obtain the following results (cf.
Prop. 3.15 - 3.19) characterizing locally projective generators which are
the counterparts of Prop. 3.7, 3.10, Theorem 3.11, Prop. 3.12, Corollary
3.13 respectively of [10] in our setting.
Proposition 3.15. RP is locally projective and Pf is finitely generated
for all f ∈ E(T ) if and only if µ : Q⊗ P → T is a surjection. Moreover,
if µ is a surjection, then it is an isomorphism.
Proof. For the necessary part, let f ∈ E(T ). Then since Pf is finitely gen-
erated, by Prop. 3.11, there exists a finitely generated projective subsemi-
module P ′ of P such that Pf ⊆ P ′, i.e.,Pf = Pf2 ⊆ P ′f ⊆ Pf . Therefore
Pf = P ′f , hence it is projective (since P ′f being a retract of P ′ is pro-
jective). Therefore by Prop. 3.7, there exists a retraction φ : (Re)n → Pf
for some n ∈ N, e2 = e ∈ R with coretraction ψ : Pf → (Re)n, i.e.,
ψφ = idPf . Consider ei ∈ (Re)n with e as the ith coordinate and all
others being 0 for each i = 1, 2, . . . , n, then for the canonical projections
πi : (Re)
n → Re we have
∑n
i=1 xπiei = x for all x ∈ (Re)n. Let pi = eiφ
and αi = πψπi for each i = 1, 2, . . . , n where π : RP →R Pf is given
by p 7→ pf . Now if we put qi = fαie ∈ T HomR(P,R)R = Q, for all
i = 1, 2, . . . , n. Then for any p ∈ P , we have pqi = p(fαie) = ((pf)αi)e =
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M. Das, S. Gupta, S. K. Sardar 47
((pf)(πψπi))e = (pf)(ψπi) for all i = 1, 2, . . . , n. Therefore for any p ∈ P ,
p
n∑
i=1
qipi =
n∑
i=1
p(qipi) =
n∑
i=1
(pqi)pi =
n∑
i=1
((pf)(ψπi))(eiφ)
= (
n∑
i=1
(pf)ψπiei)φ = (pf)ψφ = pf,
i.e., f =
∑n
i=1 qipi. Now for any t ∈ T there exists an idempotent
f =
∑n
i=1 qipi such that t = ft. Then we have t = ft =
∑n
i=1 qipit =
µ(
∑n
i=1 qi ⊗ pit). Thus µ is onto. Conversely, for any idempotent f ∈ T ,
there exist pi ∈ P, qi ∈ Q for i = 1, 2, . . . , n such that µ(
∑n
i=1 qi ⊗ pi) =∑n
i=1 qipi = f . Let e ∈ E(R) such that qie = qi for all i = 1, 2, . . . , n.
Then we define α : (Re)n → Pf by (x1, x2, . . . , xn) 7→
∑n
i=1 xipif and
β : Pf → (Re)n by y 7→ (yq1, yq2, . . . , yqn). Then for y ∈ Pf ,
yβα = (yq1, yq2, . . . , yqn)α =
n∑
i=1
(yqi)pif =
n∑
i=1
((yqi)pi)f
=
n∑
i=1
y(qipi)f = y(
n∑
i=1
qipi)f = yf2 = y,
i.e., βα = idPf . Hence Pf being a retract of (Re)n is finitely generated
projective (by Prop. 3.7). Also, P = lim−→RPf (can be proved along the
same lines as Prop. 3.10). Therefore RP is locally projective.
Now let µ be a surjection and µ(
∑m
i=1 qi ⊗ pi) = µ(
∑n
j=1 q
′
j ⊗ p′j).
Since PT is unitary there exists f ∈ E(T ) such that pif = pi, p
′
jf = p′j
for all i = 1, 2, . . . ,m and j = 1, 2, . . . , n. Now by the surjectivity of
µ, f =
∑k
l=1 ylxl, where xl ∈ P, yl ∈ Q for all l = 1, 2, . . . , k. Then we
have
m∑
i=1
qi ⊗ pi =
m∑
i=1
qi ⊗ pi(
k∑
l=1
ylxl) =
∑
i,l
qi ⊗ pi(ylxl) =
∑
i,l
qi ⊗ (piyl)xl
=
∑
i,l
qi(piyl)⊗ xl =
∑
i,l
(qipi)yl ⊗ xl =
∑
l
(
∑
i
qipi)yl ⊗ xl
=
∑
l
(
∑
j
q′jp
′
j)yl ⊗ xl = · · · =
n∑
j=1
q′j ⊗ p′j ,
which proves that µ is injective. Hence µ is an isomorphism.
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48 Morita equivalence of semirings
Proposition 3.16. RP is a generator for R-Sem if and only if τ : P⊗Q→
R is a surjection. Moreover, if τ is a surjection, then it is an isomorphism.
Proof. For the necessary part, since RP is a generator, RR is a sum
of homomorphic images of P , i.e., every r ∈ R can be written as r =∑k
i=1 piφi, pi ∈ P, φi ∈ HomR(P,R) for all i = 1, 2, . . . , k. Now, since PT
is unitary, there exists f ∈ E(T ) such that pif = pi for all i = 1, 2, . . . , k,
also there exists e ∈ E(R) such that re = r. Therefore we have
r = (
k∑
i=1
piφi)e =
k∑
i=1
(piφi)e =
k∑
i=1
pi(φie)
=
k∑
i=1
(pif)(φie) =
k∑
i=1
pi(fφie) = τ(
k∑
i=1
pi ⊗ fφie),
where fφie ∈ T HomR(P,R)R = Q. Therefore τ is onto. Conversely, let τ
be a surjection then R =
∑
q∈Q Pq ⊆
∑
q∈HomR(P,R) Pq = tr(P ), therefore
R = tr(P ). Hence RP is a generator for R-Sem.
Now if we assume τ to be surjective, then the injectivity of τ can be
proved in a manner similar to that of µ in Prop. 3.15.
Combining the above two results we obtain the following result.
Proposition 3.17. RP is a locally projective generator and RPf is finitely
generated for all f ∈ E(T ) if and only if µ : Q⊗P → T and τ : P⊗Q→ R
are T -T and R-R isomorphisms respectively.
Proposition 3.18. Let RP be a locally projective generator for R-Sem
and RPf be finitely generated for all f ∈ E(T ). Then the following hold:
(1) R ∼= (EndT P )R ∼= REndT Q as semirings.
(2) Q := T HomR(P,R)R ∼= HomT (P, T )R as T -R-bisemimodules.
(3) P ∼= RHomT (Q, T ) as R-T -bisemimodules.
(4) P ∼= (HomR(Q,R))T as R-T -bisemimodules.
(5) T ∼= (EndRQ)T as semirings.
Proof. (1) Consider the map σ : R → EndT P defined by σ(r)(p) := rp,
where r ∈ R, p ∈ P . For any r1, r2 ∈ R, p ∈ P, σ(r1+r2)p = (r1+r2)p =
r1p + r2p = σ(r1)(p) + σ(r2)(p) = (σ(r1) + σ(r2))p, i.e., σ(r1 + r2) =
σ(r1) + σ(r2). Also σ(r1r2)(p) = (r1r2)p = r1(r2p) = σ(r1)σ(r2)(p),
i.e., σ(r1r2) = σ(r1)σ(r2). Thus σ is a semiring morphism. Now let
σ(r1) = σ(r2) for some r1, r2 ∈ R. Therefore r1p = r2p for all p ∈ P .
Suppose e ∈ E(R) such that r1 = r1e, r2 = r2e. Now using Prop. 3.16,
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M. Das, S. Gupta, S. K. Sardar 49
there exist pk ∈ P, qk ∈ Q for k = 1, 2, . . . , n such that
∑n
k=1 pkqk = e.
Therefore, r1 = r1e = r1
∑n
k=1 pkqk =
∑n
k=1(r1pk)qk =
∑n
k=1(r2pk)qk =
r2
∑n
k=1 pkqk = r2e = r2. Hence σ is injective. Therefore identifying R
with the subsemiring σ(R) of EndT P , let ψ ∈ (EndT P )R, then there
exists an idempotent e′ =
∑m
i=1 p
′
iq
′
i ∈ R, such that ψe′ = ψ. Then for
any p ∈ P we have
ψ(p) = (ψe′)p = ψ(e′p) = ψ(
m∑
i=1
(p′iq
′
i)p) = ψ(
m∑
i=1
p′i(q
′
ip))
=
m∑
i=1
ψ(p′i)(q
′
ip) =
m∑
i=1
(ψ(p′i)q
′
i)p = σ(
m∑
i=1
(ψ(p′i)q
′
i))(p),
i.e., ψ = σ(
∑m
i=1(ψ(p
′
i)q
′
i)). Thus R ∼= (EndT P )R as semirings. Similarly,
considering the map ξ : R → EndT Q defined by ξ(r)(q) := qr we can
show that R ∼= REndT Q as semirings.
(2) Define the map λ : Q → HomT (P, T )R by λ(q)(p) := qp, where
q ∈ Q, p ∈ P . For q ∈ Q there exists e′ ∈ E(R) such that qe′ = q,
therefore using the QRP -associativity (λ(q)e′)p = λ(q)(e′p) = q(e′p) =
(qe′)p = qp = λ(q)(p), i.e., λ(q) = λ(q)e′ ∈ HomT (P, T )R. That λ is a
monoid morphism follows from the fact that µ is a monoid morphism and
using the QRP -associativity we have (tλ(q)r)(p) = tλ(q)(rp) = t(q(rp)) =
t((qr)p) = (tqr)p = λ(tqr)(p). Thus λ is a T -R morphism. For q, q′ ∈ Q,
let λ(q) = λ(q′) then for any p ∈ P, qp = q′p. Suppose e2 = e ∈ R
such that q = qe, q′ = q′e. Now, in view of Prop. 3.16, there exist
pk ∈ P, qk ∈ Q for k = 1, 2, . . . , n such that
∑n
k=1 pkqk = e. Therefore,
q = qe = q
∑n
k=1 pkqk =
∑n
k=1(qpk)qk =
∑n
k=1(q
′pk)qk = q′
∑n
k=1 pkqk =
q′e = q′. Let φ ∈ HomT (P, T )R, then there exists e′ =
∑m
i=1 p
′
iq
′
i ∈ E(R),
such that φe′ = φ. Then using TQP -associativity, for any p ∈ P we have
φ(p) = (φe′)p = φ(e′p) = φ(
m∑
i=1
(p′iq
′
i)p) = φ(
m∑
i=1
p′i(q
′
ip))
=
m∑
i=1
φ(p′i)(q
′
ip) =
m∑
i=1
(φ(p′i)q
′
i)p = λ(
m∑
i=1
(φ(p′i)q
′
i))(p),
i.e., φ = λ(
∑m
i=1(φ(p
′
i)q
′
i)). Thus λ is an isomorphism.
(3),(4) can be proved in a manner similar to (2) and (5) can be proved
along the same lines as (1).
Proposition 3.19. Let RP be a locally projective generator for R-Sem
and RPf be finitely generated for all f ∈ E(T ). Then TQ ∈ T -Sem, PT ∈
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50 Morita equivalence of semirings
Sem-T , QR ∈ Sem-R are locally projective generators for their respective
categories.
Proof. Suppose that RP ∈ R-Sem is a locally projective generator for
R-Sem and RPf is finitely generated for all f2 = f ∈ T . Then by
Prop. 3.18, identifying P with RHomT (Q, T ) and R with REndT Q and
using the fact that τ and µ are isomorphisms (Prop. 3.17) and finally
applying Prop. 3.17 to TQ, we have that TQ ∈ T -Sem is a locally projec-
tive generator. Similarly PT , QR can be proved to be locally projective
generators for their respective categories.
4. Morita equivalence and Morita context
Definition 4.1. Let R,S be two semirings with local units. We call R
and S to be Morita equivalent if the categories R-Sem and S-Sem are
equivalent, i.e., there exist additive functors F : R-Sem → S-Sem and
G : S-Sem → R-Sem such that F and G are mutually inverse equivalence
functors.
In what follows by equivalence functors we mean additive equivalence
functors. In this section we are going to characterize Morita equivalence
for semirings with local units (cf. Theorem 4.13). In order to achieve this
we first obtain some results below.
Definition 4.2. A unitary bisemimodule RPS is said to be faithfully
balanced if the canonical homomorphisms S → EndR P and R→ EndS P
given by s 7→ ρs(p 7→ ps) and r 7→ λr(p 7→ rp) respectively, where
s ∈ S, r ∈ R, p ∈ P , are injective and identifying R and S with the
corresponding subsemirings of endomorphisms of P , S EndR P = S and
(EndS P )R = R.
The following result is analogous to the case of categories of semimod-
ules over a semiring with identity [10] and can be proved in a similar
manner.
Lemma 4.3. Let F : R-Sem ⇄ S-Sem : G be an equivalence of the
categories R-Sem and S-Sem, and θ be a surjection in R-Sem. Then F (θ)
is a surjection in S-Sem.
Lemma 4.4. Let F : R-Sem ⇄ S-Sem : G be an equivalence of the
categories R-Sem and S-Sem, and RP ∈ R-Sem be projective. Then
F (P ) ∈ S-Sem is projective, too.
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M. Das, S. Gupta, S. K. Sardar 51
Proof. By Lemma 4.3, F preserves surjections. So in view of the definition
of projectivity in the category of unitary semimodules [cf. Def. 3.1], with
relevant modification of the proof of [9, Proposition 5.1.34] by replacing
the notion of epimorphism by surjectivity, the result follows easily.
Lemma 4.5. Let F : R-Sem ⇄ S-Sem : G be an equivalence of the
categories R-Sem and S-Sem, and RP ∈ R-Sem be a generator for R-Sem.
Then F (P ) ∈ S-Sem is a generator for S-Sem.
Proof. Let N ∈ S-Sem. Since P is a generator, there exists a surjection
α : P (I) → G(N) for some non-empty index set I. By Lemma 4.3, F (α) :
F (P (I)) → FG(N) is a surjection where FG(N) ∼= N . Also F and G
being mutually inverse equivalence functors, by [9, Prop. 5.1.31], G is the
right adjoint of F . Then by the dual of [12, Theorem 5.5.1], F preseves
direct limits, hence preserves coproducts, i.e., F (P (I)) ∼= F (P )(I). Thus
N is a homomorphic image of a direct sum of copies of F (P ). Hence F (P )
is a generator for S-Sem.
We skip the proof of Lemma 4.6 and Lemma 4.7 as they can be proved
along the same lines as in the case of module theory [3].
Lemma 4.6. Let F : R-Sem → S-Sem be a categorical equivalence.
Then for each M,M ′ ∈ R-Sem the restriction of F to HomR(M,M ′),
F : HomR(M,M ′) → HomS(F (M), F (M ′)) is a monoid isomorphism. In
particular F : EndR(M) → EndS(F (M)) is a semiring isomorphism.
Lemma 4.7. Let F : R-Sem → S-Sem be an equivalence of the categories
R-Sem and S-Sem, and let RP ∈ R-Sem be finitely generated. Then
F (P ) ∈ S-Sem is finitely generated, too.
Theorem 4.8. Let F : R-Sem ⇄ S-Sem : G be an equivalence of the
categories R-Sem and S-Sem, and let RP ∈ R-Sem be a locally projective
generator. Then F (P ) ∈ S-Sem is a locally projective generator, too.
Proof. By [9, Prop. 5.1.31], G is the right adjoint of F . Then by the dual
of [12, Theorem 5.5.1], F preseves direct limits. Using this fact together
with Lemmas 4.4, 4.5 and 4.7 we obtain the result.
In the following proposition we observe the adjointness of the tensor
functor and Hom functor between the categories of unitary semimodules.
It is a routine verification so we omit the proof.
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52 Morita equivalence of semirings
Proposition 4.9. Let R,S be semirings with local units and SAR ∈
S-Sem-R, RB ∈ R-Sem, SC ∈ S-Sem. Then
φ : HomS(A⊗B,C) → HomR(B,RHomS(A,C))
given by
α 7→ α′ : B → RHomS(A,C)
b 7→ bα′ : A→ C
a 7→ (a⊗ b)α
is a bijective mapping natural in SAR, RB, SC. In particular, the functor
RHomS(A,−) is right adjoint to the functor A⊗−.
The next result is the counterpart of Theorem 4.5 of [10] in this general
setting.
Theorem 4.10. For a functor F : R-Sem → S-Sem the following state-
ments are equivalent.
(1) F has a right adjoint.
(2) F preserves direct limits.
(3) There exists a unitary S-R-bisemimodule Q such that the functors
Q⊗− : R-Sem → S-Sem and F are naturally isomorphic.
Proof. (1) ⇒ (2) and (3) ⇒ (1) follow from the right analogue of [12,
Theorem 5.5.1] and Prop. 4.9 respectively.
(2) ⇒ (3) Let Q := F (R) ∈ S-Sem. Then F induces a right R-
semimodule structure on Q with the R-action given by Q × R → Q by
(q, r) 7→ qF (ρr) where ρr : R → R is given by x 7→ xr. In order to show
that QR is unitary, suppose q ∈ Q. Now Q = F ( ∪
e∈E(R)
Re) = ∪
e∈E(R)
F (Re)
(since R is a semiring with local units, union coincides in this formula with
direct limit and by the hypothesis F preserves direct limits). Therefore
q ∈ F (Re) for some idempotent e ∈ R. Then we have qe = qF (ρe) = q
(since ρe = 1Re implies that F (ρe) = 1F (Re)). Thus Q is a unitary S-
R-bisemimodule. Then the proof follows similarly as in [10, Theorem
4.5].
Theorem 4.11. Let R and S be Morita equivalent semirings with local
units via inverse equivalences F : R-Sem → S-Sem and G : S-Sem →
R-Sem. Set P = G(S) and Q = F (R). Then the following hold:
(1) RPS , SQR are unitary faithfully balanced bisemimodules.
(2) RP, PS , SQ,QR are locally projective generators.
(3) F ∼= Q⊗−, G ∼= P ⊗−.
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M. Das, S. Gupta, S. K. Sardar 53
(4) F ∼= SHomR(P,−), G ∼= RHomS(Q,−).
(5) RPS
∼= RHomS(Q,S) ∼= (HomR(Q,R))S and
SQR
∼= SHomR(P,R) ∼= HomS(P, S)R.
Proof. LetG(S) = P , thenG being an equivalence functor using Lemma 4.6
we have EndS S ∼= EndR P as semirings. By Prop. 3.12, S ∼= S EndS S
as semirings. Since P is a right EndR P -semimodule, identifying S with
the subsemiring S EndS S of EndS S, P can be considered as a right S-
semimodule with the action P × S → P given by (p, s) 7→ pG(ρs) where
ρs : S → S is given by t 7→ ts. That PS is unitary follows similarly as in
the proof of Theorem 4.10. Thus P is a unitary R-S-bisemimodule. Now
since S is a locally projective generator, by Theorem 4.8, RP = G(S) is
a locally projective generator. In view of Lemma 4.7, Pf = G(Sf) is a
finitely generated left S-semimodule for all f ∈ E(S) and S ∼= S EndS S ∼=
S EndR P as semirings. Since RP is a locally projective generator with
Pf finitely generated for all f2 = f ∈ S, using (1) of Prop. 3.18 we
have R ∼= (EndS P )R as semirings. Hence RPS is a faithfully balanced
bisemimodule. Similarly Q = F (R) is a unitary faithfully balanced S-R-
bisemimodule. Hence (1) is proved.
Since F and G are mutually inverse equivalence functors, they are
adjoint to each other [9, Prop. 5.1.31]. Therefore using Theorem 4.10, we
obtain F ∼= Q⊗−. Similarly G ∼= P⊗−. By Prop. 4.9,Q⊗− is left adjoint
to RHomS(Q,−) and P ⊗ − is left adjoint to SHomR(P,−). Then by
uniqueness of adjoint functors upto natural isomorphism [9, Cor. 5.1.10],
we obtain F ∼= Q⊗− ∼= SHomR(P,−) and G ∼= P ⊗− ∼= RHomS(Q,−).
This proves (3) and (4).
Now using (4) we obtain, P = G(S) ∼= RHomS(Q,S) as R-S-
bisemimodule and Q = F (R) ∼= SHomR(P,R) as S-R-bisemimodule.
Since by (1), QR is unitary, using Prop. 3.18 we obtain, Q =
QR ∼= SHomR(P,R)R ∼= HomS(P, S)R as S-R-bisemimodule and also
P ∼= (HomR(Q,R))S as R-S-bisemimodule, which proves (5). Now (2)
clearly follows from Prop. 3.19.
Definition 4.12. [15] Let R and S be two semirings and RPS and SQR
be an R-S-bisemimodule and an S-R-bisemimodule, respectively and τ :
P⊗SQ→ R and µ : Q⊗RP → S be an R-S-bisemimodule homomorphism
and an S-R-bisemimodule homomorphism, respectively, such that τ(p⊗
q)p′ = pµ(q⊗p′) and µ(q⊗p)q′ = qτ(p⊗ q′) for all p, p′ ∈ P and q, q′ ∈ Q.
Then the sixtuple (R,S, U, V, τ, µ) is called a Morita context for semirings.
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54 Morita equivalence of semirings
Moreover, we say that a Morita context is unitary if RPS and SQR are
unitary bisemimodules.
Theorem 4.13. Let R and S be semirings with local units. Then the
following are equivalent:
(1) R and S are Morita equivalent.
(2) There exists a faithfully balanced unitary bisemimodule RPS such
that RP is a locally projective generator and RPf is finitely generated for
all f ∈ E(S).
(3) There exists a unitary Morita context (R,S,R PS ,S QR, τ, µ) with
surjective τ, µ.
(4) There exists a unitary Morita context (R,S,R PS ,S QR, τ, µ) with
bijective τ, µ.
Proof. (1) ⇒ (2) Let P := G(S). Then the proof follows from Theo-
rem 4.11.
(2) ⇒ (3) Suppose there exists a unitary bisemimodule RPS such that
RP is a locally projective generator and RPf is finitely generated for
all f ∈ E(S) and S ∼= S EndR P as semirings. Let Q = SHomR(P,R)R.
Then define:
τ : P ⊗Q→ R
p⊗ q 7→ pq
and µ : Q⊗ P → S
q ⊗ p 7→ qp (p′ 7→ (p′q)p)
It is routine to verify that the maps τ, µ are respectively R−R and S−S
morphisms. For any p′ ∈ P ,
p′µ(q ⊗ p)q′ = p′((qp)q′) = (p′(qp))q′ = ((p′q)p)q′
= (p′q)(pq′) = p′(q(pq′)) = p′(qτ(p⊗ q′)),
i.e., µ(q⊗p)q′ = qτ(p⊗ q′). Also τ(p⊗ q)p′ = (pq)p′ = p(qp′) = pµ(q⊗p′).
Consequently (R,S,R PS ,S QR, τ, µ) is a Morita context. By hypothesis,
RP is a locally projective generator and RPf is finitely generated for all
f ∈ E(S) and S ∼= S EndR P as semirings. Hence using Prop. 3.16 and
Prop. 3.15, we get that τ, µ are surjections.
(3) ⇒ (4) Suppose (R,S,R PS ,S QR, τ, µ) is a unitary Morita context
with surjective τ, µ. Let τ(
∑m
i=1 pi⊗ qi) = τ(
∑n
j=1 p
′
j ⊗ q′j), where pi, p′j ∈
P, qi, q
′
j ∈ Q for all i = 1, 2, . . . ,m, j = 1, 2, . . . , n. Since QR is unitary,
there exists an idempotent e ∈ R such that qie = qi, q
′
je = q′j for
all i = 1, 2, . . . ,m, j = 1, 2, . . . , n. Now by the surjectivity of τ, e =
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M. Das, S. Gupta, S. K. Sardar 55
τ(
∑k
l=1 xl ⊗ yl), where xl ∈ P, yl ∈ Q for all l = 1, 2, . . . , k. Therefore we
have,
m∑
i=1
pi ⊗ qi =
m∑
i=1
pi ⊗ qiτ(
k∑
l=1
xl ⊗ yl) =
∑
i,l
pi ⊗ qiτ(xl ⊗ yl)
=
∑
i,l
pi ⊗ µ(qi ⊗ xl)yl =
∑
i,l
piµ(qi ⊗ xl)⊗ yl
=
∑
l
∑
i
τ(pi ⊗ qi)xl ⊗ yl =
k∑
l=1
τ(
m∑
i=1
pi ⊗ qi)xl ⊗ yl
=
k∑
l=1
τ(
n∑
j=1
p′j ⊗ q′j)xl ⊗ yl = · · · =
n∑
j=1
p′j ⊗ q′j ,
which proves that τ is injective. Similarly µ is also injective.
(4) ⇒ (1) Let (R,S,R PS ,S QR, τ, µ) be a unitary Morita context with
bijective τ, µ. Then P ⊗S Q ∼= R and Q ⊗R P ∼= S. Therefore for every
M ∈ R-Sem, P ⊗S (Q ⊗R M) ∼= (P ⊗S Q) ⊗R M ∼= R ⊗R M ∼= M
(cf. Prop. 3.14). Now we consider the class of isomorphisms η = {ηX :
P ⊗S (Q⊗R X) →R X| X ∈ R-Sem}. Then η is a natural isomorphism
between the identity functor 1R-Sem on the category R-Sem and the functor
P ⊗S (Q ⊗R −) as for all RX,R Y ∈ R-Sem and f ∈ HomR(X,Y ) the
following diagram commutes:
X
f // Y
R⊗X
OO
1R⊗f // R⊗ Y
OO
P ⊗Q⊗X
OO
1P⊗1Q⊗f
// P ⊗Q⊗ Y
OO
Then P ⊗S (Q⊗R −) ∼= 1R-Sem. Similarly Q⊗R (P ⊗S −) ∼= 1S-Sem. Thus
P ⊗S − : S-Sem → R-Sem : Q ⊗R − is an equivalence of the categories
R-Sem and S-Sem.
Analogously to Corollary 4.3 of [1], we have the following proposition.
Proposition 4.14. Let R be a semiring with slu. Then the following are
equivalent:
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56 Morita equivalence of semirings
(1) R is Morita equivalent to a semiring with identity.
(2) There exists an idempotent e ∈ R such that R = ReR.
Proof. (1) ⇒ (2) Suppose R is Morita equivalent to a semiring S with
identity via inverse equivalences F : R-Sem ⇄ S-Sem : G. Let P = G(S).
Since S is a finitely generated projective generator, RP also is a finitely gen-
erated projective generator. Now RP being a finitely generated projective
unitary R-semimodule, by Prop. 3.7, there exists a surjective R-morphism
φ : (Re)m → P for some idempotent e ∈ R and m ∈ N which implies that
Re is a generator for R-Sem. Also since for any r ∈ R, Rr is finitely gener-
ated, using Prop. 3.4 there exists a surjective R-morphism ψ : (Re)n → Rr
for some n ∈ N. Therefore there exists (r1, r2, . . . , rn) ∈ (Re)n such that
r = (r1, r2, . . . , rn)ψ = r1e((e, 0, . . . , 0)ψ) + · · · + rne((0, . . . , 0, e)ψ) ∈
ReRr ⊆ ReR, which is true for any r ∈ R. Therefore R = ReR.
(2) ⇒ (1) Let P = Re. Then clearly P is a finitely generated projective
unitary R-semimodule. Also for any M ∈ R-Sem, for each m ∈M consider
the map ρm : P → M defined by y 7→ ym, where y ∈ P , m ∈ M . Then
ρ =
⊕
m∈M ρm :
⊕
M P →M , where (
⊕
M P )ρ =
∑
m∈M Pρm = PM =
P (RM) = (PR)M = (ReR)M = RM = M , which implies that ρ is a
surjection. Thus P is a finitely generated projective generator hence a
locally projective generator for R-Sem. Now if we take S = EndR P =
EndR(Re) = eRe, then using (2) of Theorem 4.13, R and S = eRe are
Morita equivalent semirings.
5. Morita invariant properties
In this section we discuss some properties of semirings with local units
which remain invariant under Morita equivalence. The results obtained
here are nothing but counterparts of the results of [17] in the setting of
semirings with local units.
Theorem 5.1. Let R and S be Morita equivalent semirings with local
units via the Morita context (R,S,RPS , SQR, τ, µ). Then R is additively
cancellative (additively idempotent, additively regular, zero-sum free) if
and only if P is additively cancellative (respectively additively idempotent,
additively regular, zero-sum free).
Proof. Let R be additively cancellative and a, b, c ∈ P such that a+ c =
b + c. Also let t ∈ S be an idempotent such that a = at, b = bt. Then
the result can be proved in a similar manner to that of [17, Theorem
2.1] by replacing t in place of 1S . Other parts follow similarly from their
corresponding definitions.
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M. Das, S. Gupta, S. K. Sardar 57
Theorem 5.2. Let R and S be Morita equivalent semirings with local
units via the Morita context (R,S,R PS ,S QR, τ, µ). Then the lattice Id(R)
of ideals of R and the lattice Sub(P ) of subsemimodules of P are isomor-
phic. Moreover, the isomorphism takes finitely generated ideals to finitely
generated subsemimodules and vice-versa.
Proof. Let us define
f : Id(R) → Sub(P ) and g : Sub(P ) → Id(R)
by
f(I) := {
n∑
k=1
ikpk | pk ∈ P, ik ∈ I for all k; n ∈ N},
and
g(N) := {
n∑
k=1
τ(pk ⊗ qk) | pk ∈ N, qk ∈ Q for all k; n ∈ N},
respectively. Then with relevant modification of the proof of Theorem 2.2
[17] by replacing the identity by a local unit the rest of the proof can be
completed.
Remark 5.3. The above result has its counterpart for k-ideals and h-
ideals which is analogous to Theorem 2.5 of [17].
Remark 5.4. f and g also preserve k-closure and h-closure.
The following result is an obvious corollary of Theorem 5.2 and the
result mentioned in Remark 5.3.
Corollary 5.5. Let R and S be Morita equivalent semirings with local
units via the Morita context (R,S,R PS ,S QR, τ, µ). Then R is ideal-simple
(k-ideal simple, h-ideal simple) if and only if P is subsemimodule-simple
(respectively k-subsemimodule simple, h-subsemimodule simple).
The following result is the counterpart of Theorem 2.8 of [17] in the
present setting.
Theorem 5.6. Let R and S be Morita equivalent semirings with local
units via the Morita context (R,S,RPS , SQR, τ, µ). Then R is Noetherian
if and only if P is Noetherian.
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58 Morita equivalence of semirings
Theorem 5.7. Let R and S be Morita equivalent semirings with local units
via the Morita context (R,S,R PS ,S QR, τ, µ). Then the lattices Con(R)
and Con(P ) of congruences of R and P respectively are isomorphic. More-
over the isomorphism takes Bourne congruences to Bourne congruences,
Iizuka congruences to Iizuka congruences and ring congruences to module
congruences and vice-versa.
Proof. Let us define
α : Con(R) → Con(P ) by α(ρ) := αρ
tr
and
β : Con(P ) → Con(R) by β(σ) := βσ
tr,
where
αρ = {(
n∑
k=1
rkpk,
n∑
k=1
r′kpk) | (rk, r
′
k) ∈ ρ, pk ∈ P for all k; n ∈ N}
and
βσ = {(
n∑
k=1
τ(pk⊗qk),
n∑
k=1
τ(p′k⊗qk)) | (pk, p
′
k)∈σ, qk∈Q for all k; n∈N}.
The rest of the proof is a slight modification of the proof of Theorem 2.10
of [17].
The following result is an obvious corollary of the above theorem.
Corollary 5.8. Let R and S be Morita equivalent semirings with local
units via the Morita context (R,S,R PS ,S QR, τ, µ). Then R is (Bourne,
Iizuka, ring) congruence-simple if and only if P is (Bourne, Iizuka, module)
congruence-simple.
6. Concluding remark
All the above results in section 5 investigate relationship between R
and P . But similar relationship can be established between R and Q, S
and P , S and Q i.e., Theorems 5.1, 5.2, 5.6, 5.7 have their counterparts
for other pairs of the components of Morita equivalent semirings with
local units. Since a semiring with identity is also a semiring with local
units, Theorems 4.11, 4.13 include some of the results of Theorems 4.6,
4.8 of [15].
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M. Das, S. Gupta, S. K. Sardar 59
Acknowledgement
The authors are very much thankful to the learned referee for his
meticulous referring and subsequent detection of some errors both in
mathematics as well as presentation whose compliance has made the
paper what it is now. The authors are also grateful to Prof. L. Márki and
Prof. P. N. Ánh of Alfréd Rényi Institute of Mathematics, Hungary for a
necessary discussion in order to resolve one issue raised by the learned
referee. Lastly the authors would like to acknowledge the encouragement
provided by Prof. Y. Katsov of Hanover College, USA at the time of
preparation of the paper.
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Contact information
Monali Das,
Sujit Kumar
Sardar
Department of Mathematics,
Jadavpur University, Kolkata, India
E-Mail(s): monali.ju7@gmail.com,
sksardarjumath@gmail.com,
sujitk.sardar@jadavpuruniversity.in
Sugato Gupta Department of Mathematics,
Vidyasagar College for Women, Kolkata,
India
E-Mail(s): sguptaju@gmail.com
Received by the editors: 16.11.2018
and in final form 02.09.2019.
mailto:monali.ju7@gmail.com
mailto:sksardarjumath@gmail.com
mailto:sujitk.sardar@jadavpuruniversity.in
mailto:sguptaju@gmail.com
M. Das, S. Gupta, S. K. Sardar
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