On radical square zero rings
Let Λ be a connected left artinian ring with radical square zero and with n simple modules. If Λ is not self-injective, then we show that any module M with Exti(M, Λ) = 0 for 1 ≤ i ≤ n + 1 is projective. We also determine the structure of the artin algebras with radical square zero and n simple modu...
Збережено в:
| Опубліковано в: : | Algebra and Discrete Mathematics |
|---|---|
| Дата: | 2012 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2012
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/152245 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On radical square zero rings / C.M. Ringel, B.-L. Xiong // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 297–306. — Бібліогр.: 4 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859478411709251584 |
|---|---|
| author | Ringel, C.M. Xiong, B.-L. |
| author_facet | Ringel, C.M. Xiong, B.-L. |
| citation_txt | On radical square zero rings / C.M. Ringel, B.-L. Xiong // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 297–306. — Бібліогр.: 4 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | Let Λ be a connected left artinian ring with radical square zero and with n simple modules. If Λ is not self-injective, then we show that any module M with Exti(M, Λ) = 0 for 1 ≤ i ≤ n + 1 is projective. We also determine the structure of the artin algebras with radical square zero and n simple modules which have a non-projective module M such that Exti(M, Λ) = 0 for 1 ≤ i ≤ n.
|
| first_indexed | 2025-11-24T11:44:42Z |
| format | Article |
| fulltext |
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 14 (2012). Number 2. pp. 297 – 306
c© Journal “Algebra and Discrete Mathematics”
On radical square zero rings
Claus Michael Ringel and Bao-Lin Xiong
Communicated by Yu. A. Drozd
Dedicated to the memory of Andrei Roiter
Abstract. Let Λ be a connected left artinian ring with
radical square zero and with n simple modules. If Λ is not self-
injective, then we show that any module M with Exti(M, Λ) = 0
for 1 ≤ i ≤ n + 1 is projective. We also determine the structure of
the artin algebras with radical square zero and n simple modules
which have a non-projective module M such that Exti(M, Λ) = 0
for 1 ≤ i ≤ n.
Xiao-Wu Chen [C] has recently shown: given a connected artin algebra
Λ with radical square zero then either Λ is self-injective or else any CM
module is projective. Here we extend this result by showing: If Λ is a
connected artin algebra with radical square zero and n simple modules
then either Λ is self-injective or else any module M with Exti(M, Λ) = 0
for 1 ≤ i ≤ n + 1 is projective. Actually, we will not need the assumption
on Λ to be an artin algebra; it is sufficient to assume that Λ is a left
artinian ring. And we show that for artin algebras the bound n + 1 is
optimal by determining the structure of those artin algebras with radical
square zero and n simple modules which have a non-projective module
M such that Exti(M, Λ) = 0 for 1 ≤ i ≤ n.
From now on, let Λ be a left artinian ring with radical square zero,
this means that Λ has an ideal I with I2 = 0 (the radical) such that
2010 MSC: 16D90, 16G10; 16G70.
Key words and phrases: Artin algebras; left artinian rings; representations,
modules; Gorenstein modules, CM modules; self-injective algebras; radical square zero
algebras.
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.
298 On radical square zero rings
Λ/I is semisimple artinian. We also assume that Λ is connected (the
only central idempotents are 0 and 1). The modules to be considered
are usually finitely generated left Λ-modules. Let n be the number of
(isomorphism classes of) simple modules.
Given a module M , we denote by PM a projective cover, by QM an
injective envelope of M . Also, we denote by ΩM a syzygy module for M ,
this is the kernel of a projective cover PM → M. Since Λ is a ring with
radical square zero, all the syzygy modules are semisimple. Inductively,
we define Ω0M = M, and Ωi+1M = Ω(ΩiM) for i ≥ 0.
Lemma 1. If M is a non-projective module with Exti(M, Λ) = 0 for
1 ≤ i ≤ d + 1 (and d ≥ 1), then there exists a simple non-projective
module S with Exti(S, Λ) = 0 for 1 ≤ i ≤ d.
Proof. The proof is obvious: We have Exti(M, Λ) ≃ Exti−1(ΩM, Λ), for
all i ≥ 2. Since M is not projective, ΩM 6= 0. Now ΩM is semisimple. If all
simple direct summands of ΩM are projective, then also ΩM is projective,
but then the condition Ext1(M, Λ) = 0 implies that Ext1(M, ΩM) = 0 in
contrast to the existence of the exact sequence 0 → ΩM → PM → M →
0. Thus, let S be a non-projective simple direct summand of ΩM.
Lemma 2. If S is a non-projective simple module with Ext1(S, Λ) = 0,
then PS is injective and ΩS is simple and not projective.
Proof. First, we show that PS has length 2. Otherwise, ΩS is of length
at least 2, thus there is a proper decomposition ΩS = U ⊕ U ′ and then
there is a canonical exact sequence
0 → PS → PS/U ⊕ PS/U ′ → S → 0,
which of course does not split. But since Ext1(S, Λ) = 0, we have
Ext1(S, P ) = 0, for any projective module P . Thus, we obtain a con-
tradiction.
This shows also that ΩS is simple. Of course, ΩS cannot be projective,
again according to the assumption that Ext1(S, P ) = 0, for any projective
module P .
Now let us consider the injective envelope Q of ΩS. It contains PS as
a submodule (since PS has ΩS as socle). Assume that Q is of length at
least 3. Take a submodule I of Q of length 2 which is different from PS
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.
C. M. Ringel, B.-L. Xiong 299
and let V = PS + I, this is a submodule of Q of length 3. Thus, there
are the following inclusion maps u1, u2, v1, v2:
ΩS
u1
−−−−→ PS
v1
y
y
u2
I
v2
−−−−→ V
The projective cover p : PI → I has as restriction a surjective map
p′ : rad PI → ΩS. But rad PI is semisimple, thus p′ is a split epimorphism,
thus we obtain a map w : ΩS → PI such that pw = v1. We consider the
exact sequence induced from the sequence 0 → ΩS → PS → S → 0 by
the map w:
0 −−−−→ ΩS
u1
−−−−→ PS
e1
−−−−→ S −−−−→ 0
w
y
yw′
∥
∥
∥
0 −−−−→ PI
u′
1
−−−−→ N
e′
1
−−−−→ S −−−−→ 0
Here, N is the pushout of the two maps u1 and w. Since we know that
u2u1 = v2v1 = v2pw, there is a map f : N → V such that fu′
1 = v2p and
fw′ = u2. Since the map
[
v2p u2
]
: PI ⊕ PS → V is surjective, also f
is surjective.
But recall that we assume that Ext1(S, Λ) = 0, thus Ext1(S, PI) = 0.
This means that the lower exact sequence splits and therefore the socle
of N = PI ⊕ S is a maximal submodule of N (since I is a local module,
also PI is a local module). Now f maps the socle of N into the socle of
V , thus it maps a maximal submodule of N into a simple submodule of
V . This implies that the image of f has length at most 2, thus f cannot
be surjective. This contradiction shows that Q has to be of length 2, thus
Q = PS and therefore PS is injective.
Lemma 3. If S is a non-projective simple module with Exti(S, Λ) = 0
for 1 ≤ i ≤ d, then the modules Si = ΩiS with 0 ≤ i ≤ d are simple and
not projective, and the modules P (Si) are injective for 0 ≤ i < d.
Proof. The proof is again is obvious, we use induction. If d ≥ 2, we know
by induction that the modules Si with 0 ≤ i ≤ d − 1 are simple and not
projective, and that the modules P (Si) are injective for 0 ≤ i < d − 1.
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.
300 On radical square zero rings
But Ext1(Ωd−1S, Λ) ≃ Extd(S, Λ) = 0, thus Lemma 2 asserts that also
Sd is simple and not projective and that P (Sd−1) is injective.
Lemma 4. Let S0, S1, . . . , Sb be simple modules with Si = Ωi(S0) for
1 ≤ i ≤ b. Assume that there is an integer 0 ≤ a < b such that the
modules Si with a ≤ i < b are pairwise non-isomorphic, whereas Sb is
isomorphic to Sa. In addition, we asssume that the modules P (Si) for
a ≤ i < b are injective. Then Sa, . . . , Sb−1 is the list of all the simple
modules and Λ is self-injective.
Proof. Let S be the subcategory of all modules with composition factors
of the form Si, where a ≤ i < b. We claim that this subcategory is closed
under projective covers and injective envelopes. Indeed, the projective
cover of Si for a ≤ i < b has the composition factors Si and Si+1 (and
Sb = Sa), thus is in S. Similarly, the injective envelope for Si with
a < i < b is Q(Si) = P (Si−1), thus it has the composition factors Si−1
and Si, and Q(Sa) = Q(Sb) = P (Sb−1) has the composition factors Sb−1
and Sa. Since we assume that Λ is connected, we know that the only non-
trivial subcategory closed under composition factors, extensions, projective
covers and injective envelopes is the module category itself. This shows
that Sa, . . . , Sb−1 are all the simple modules. Since the projective cover
of any simple module is injective, Λ is self-injective.
Theorem 1. Let Λ be a connected left artinian ring with radical square
zero. Assume that Λ is not self-injective. If S is a non-projective simple
module such that Exti(S, Λ) = 0 for 1 ≤ i ≤ d, then the modules Si = ΩiS
with 0 ≤ i ≤ d are pairwise non-isomorphic simple and non-projective
modules and the modules P (Si) are injective for 0 ≤ i < d.
Proof. According to Lemma 3, the modules Si (with 0 ≤ i ≤ d) are simple
and non-projective, and the modules P (Si) are injective for 0 ≤ i < d.
If at least two of the modules S0, . . . , Sd are isomorphic, then Lemma 4
asserts that Λ is self-injective, but this we have excluded.
Theorem 2. Let Λ be a connected left artinian ring with radical square
zero and with n simple modules. The following conditions are equivalent:
(i) Λ is self-injective, but not a simple ring.
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.
C. M. Ringel, B.-L. Xiong 301
(ii) There exists a non-projective module M with Exti(M, Λ) = 0 for
1 ≤ i ≤ n + 1.
(iii) There exists a non-projective simple module S with Exti(S, Λ) = 0
for 1 ≤ i ≤ n.
Proof. First, assume that Λ is self-injective, but not simple. Since Λ is not
semisimple, there is a non-projective module M . Since Λ is self-injective,
Exti(M, Λ) = 0 for all i ≥ 1. This shows the implication (i) =⇒ (ii).
The implication (ii) =⇒ (iii) follows from Lemma 1. Finally, for the
implication (iii) =⇒ (i) we use Theorem 1. Namely, if Λ is not self-
injective, then Theorem 1 asserts that the simple modules Si = ΩiS with
0 ≤ i ≤ n are pairwise non-isomorphic. However, these are n + 1 simple
modules, and we assume that the number of isomorphism classes of simple
modules is n. This completes the proof of Theorem 2.
Note that the implication (ii) =⇒ (i) in Theorem 2 asserts in
particular that either Λ is self-injective or else that any CM module is
projective, as shown by Chen [C]. Let us recall that a module M is said to
be a CM module provided Exti(M, Λ) = 0 and Exti(Tr M, Λ) = 0, for all
i ≥ 1 (here Tr denotes the transpose of the module); these modules are
also called Gorenstein-projective modules, or totally reflexive modules, or
modules of G-dimension equal to 0. Note that in general there do exist
modules M with Exti(M, Λ) = 0 for all i ≥ 1 which are not CM modules,
see [JS].
We also draw the attention to the generalized Nakayama conjecture
formulated by Auslander-Reiten [AR]. It asserts that for any artin algebra
Λ and any simple Λ-module S there should exist an integer i ≥ 0 such that
Exti(S, Λ) 6= 0. It is known that this conjecture holds true for algebras
with radical square zero. The implication (iii) =⇒ (i) of Theorem 2
provides an effective bound: If n is the number of simple Λ-modules, and
S is simple, then Exti(S, Λ) 6= 0 for some 0 ≤ i ≤ n. Namely, in case S
is projective or Λ is self-injective, then Ext0(S, Λ) 6= 0. Now assume that
S is simple and not projective and that Λ is not self-injective. Then there
must exist some integer 1 ≤ i ≤ n with Exti(S, Λ) 6= 0, since otherwise
the condition (iii) would be satisfied and therefore condition (i).
Theorem 1 may be interpreted as a statement concerning the Ext-
quiver of Λ. Recall that the Ext-quiver Γ(R) of a left artinian ring R has
as vertices the (isomorphism classes of the) simple R-modules, and if S, T
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.
302 On radical square zero rings
are simple R-modules, there is an arrow T → S provided Ext1(T, S) 6= 0,
thus provided that there exists an indecomposable R-module M of length
2 with socle S and top T . We may add to the arrow α : T → S the
number l(α) = ab, where a is the length of soc PT and b is the length
of QS/ soc (note that b may be infinite). The properties of Γ(R) which
are relevant for this note are the following: the vertex S is a sink if and
only if S is projective; the vertex S is a source if and only if S is injective;
finally, if R is a radical square zero ring and S, T are simple R-modules
then PT = QS if and only if there is an arrow α : T → S with l(α) = 1
and this is the only arrow starting at T and the only arrow ending in S.
Theorem 1 assert the following: Let Λ be a connected left artinian
ring with radical square zero. Assume that Λ is not self-injective. Let S
be a non-projective simple module such that Exti(S, Λ) = 0 for 1 ≤ i ≤ d,
and let Si = ΩiS with 0 ≤ i ≤ d. Then the local structure of Γ(Λ) is as
follows:
S0 S1 Sd−1 Sd
. . ..............................................................................
.....
...
..
..
..................................................................
.....
...
..
..
................................................................................
....
.....
..
..
..
..
..
..
................................................................................. ..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.........
..
..
............
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.......
...
..
.
............
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...............
............
................................................................................
....
.....
..
..
..
.
..
..
.
.
.
.
.
.
.
1 1
such that there is at least one arrow starting in Sd (but maybe no arrow
ending in S0). To be precise: the picture is supposed to show all the
arrows starting or ending in the vertices S0, . . . , Sd (and to assert that
the vertices S0, . . . , Sd are pairwise different).
Let us introduce the quivers ∆(n, t), where n, t are positive integers.
The quiver ∆(n, t) has n vertices and also n arrows, namely the vertices
labeled 0, 1, . . . , n − 1, and arrows i → i+1 for 0 ≤ i ≤ n − 1 (modulo n)
(thus, we deal with an oriented cycle); in addition, let l(α) = t for the
arrow α : n − 1 → 0 and let l(β) = 1 for the remaining arrows β:
◦
◦
◦
◦
◦
◦
.....................................................................
..
.
..
..
.
..
..
...
....
...
.....................................................................
..
...
.....
..
..
..
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.............
....
...
.
..
.
.
.........................................................
...
..
..
............
................................................................
.....
...
..
..
.
.
.
.
.
.
.
0
1
2 3
n−2
n−1
1
1
1
1
t
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.
C. M. Ringel, B.-L. Xiong 303
Note that the Ext-quiver of a connected self-injective left artinian ring
with radical square zero and n vertices is just ∆(n, 1). Our further interest
lies in the cases t > 1.
Theorem 3. Let Λ be a connected left artinian ring with radical square
zero and with n simple modules.
(a) If there exists a non-projective simple modules S with Exti(S, Λ) = 0
for 1 ≤ i ≤ n − 1, or if there exists a non-projective module M with
Exti(M, Λ) = 0 for 1 ≤ i ≤ n, then Γ(Λ) is of the form ∆(n, t) with
t > 1.
(b) Conversely, if Γ(Λ) = ∆(n, t) and t > 1, then there exists a unique
simple module S with Exti(S, Λ) = 0 for 1 ≤ i ≤ n − 1, namely the
module S = S(0) (and it satisfies Extn(S, Λ) 6= 0).
(c) If Γ(Λ) = ∆(n, t) and t > 1, and if we assume in addition that Λ is
an artin algebra, then there exists a unique indecomposable module
M with Exti(M, Λ) = 0 for 1 ≤ i ≤ n, namely M = Tr D(S(0))
(and it satisfies Extn+1(M, Λ) 6= 0).
Here, for Λ an artin algebra, D denotes the k-duality, where k is the center
of Λ (thus D = Homk(−, E), where E is a minimal injective cogenerator in
the category of k-modules); thus D Tr is the Auslander-Reiten translation
and Tr D the reverse.
Proof of Theorem 3. Part (a) is a direct consequence of Theorem 1, using
the interpretation in terms of the Ext-quiver as outlined above. Note that
we must have t > 1, since otherwise Λ would be self-injective.
(b) We assume that Γ(Λ) = ∆(n, t) with t > 1. For 0 ≤ i < n, let
S(i) be the simple module corresponding to the vertex i, let P (i) be its
projective cover, I(i) its injective envelope. We see from the quiver that
all the projective modules P (i) with 0 ≤ i ≤ n − 2 are injective, thus
Extj(−, Λ) = Extj(−, P (n−1)) for all j ≥ 1. In addition, the quiver shows
that ΩS(i) = S(i+1) for 0 ≤ i ≤ n−2. Finally, we have ΩS(n−1) = S(0)a
for some positive integer a dividing t and the injective envelope of P (n−1)
yields an exact sequence
0 → P (n − 1) → I(P (n − 1)) → S(n − 1)t−1 → 0 (*)
(namely, I(P (n − 1)) = I(soc P (n − 1)) = I(S(0)a) = I(S(0))a and
I(S(0))/ soc is the direct sum of b copies of S(n − 1), where ab = t; thus
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.
304 On radical square zero rings
the cokernel of the inclusion map P (n − 1) → I(P (n − 1)) consists of t − 1
copies of S(n − 1)).
Since t > 1, the exact sequence (∗) shows that Ext1(S(n − 1),
P (n − 1)) 6= 0. It also implies that Ext1(S(i), P (n − 1)) = 0
for 0 ≤ i ≤ n − 2, and therefore that
Exti(S(0), P (n − 1)) = Ext1(Ωi−1S(0), P (n − 1))
= Ext1(S(i − 1), P (n − 1))
= 0
for 1 ≤ i ≤ n − 1.
Since Ωn−i−1S(i) = S(n − 1) for 0 ≤ i ≤ n − 1, we see that
Extn−i(S(i), P (n − 1)) = Ext1(Ωn−i−1S(i), P (n − 1))
= Ext1(S(n − 1), P (n − 1))
6= 0
for 0 ≤ i ≤ n − 1. Thus, on the one hand, we have Extn(S(0), Λ) 6= 0,
this concludes the proof that S(0) has the required properties. On the
other hand, we also see that S = S(0) is the only simple module with
Exti(S, Λ) = 0 for 1 ≤ i ≤ n − 1. This completes the proof of (b).
(c) Assume now in addition that Λ is an artin algebra. As usual, we
denote the Auslander-Reiten translation D Tr by τ. Let M be a non-
projective indecomposable module with Exti(M, Λ) = 0 for 1 ≤ i ≤ n.
The shape of Γ(Λ) shows that ΩM = Sc for some simple module S (and
we have c ≥ 1), also it shows that no simple module is projective. Now
Exti(S, Λ) = 0 for 1 ≤ i < n, thus according to (b) we must have S = S(0).
It follows that PM has to be a direct sum of copies of P (n − 1), say of d
copies. Thus a minimal projective presentation of M is of the form
P (0)c → P (n − 1)d → M → 0,
and therefore a minimal injective copresentation of τM is of the form
0 → τM → I(0)c → I(n − 1)d.
In particular, soc τM = S(0)c and (τM)/ soc is a direct sum of copies of
S(n − 1).
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.
C. M. Ringel, B.-L. Xiong 305
Assume that τM 6= S(0), thus it has at least one composition factor of
the form S(n−1) and therefore there exists a non-zero map f : P (n−1) →
τM. Since τM is indecomposable and not injective, any map from an
injective module to τM maps into the socle of τM . But the image of f is
not contained in the socle of τM , therefore f cannot be factored through
an injective module. It follows that
Ext1(M, P (n − 1)) ≃ DHom(P (n − 1), τM) 6= 0,
which contradicts the assumption that Ext1(M, Λ) = 0. This shows that
τM = S(0) and therefore M = Tr DS(0).
Of course, conversely we see that M = Tr DS(0) satisfies
Exti(M, P (n − 1)) = 0 for 1 ≤ i ≤ n, and Extn+1(M, P (n − 1)) 6= 0.
Remarks.
(1) The module M = Tr DS(0) considered in (c) has length t2 + t − 1,
thus the number t (and therefore ∆(n, t)) is determined by M .
(2) If Λ is an artin algebra with Ext-quiver ∆(n, t), the number t has
to be the square of an integer, say t = m2. A typical example of
such an artin algebra is the path algebra of the following quiver
◦
◦
◦
◦
◦
◦
.....................................................................
.
..
.
..
..
.
.
..
..
...
.....
.....................................................................
...
....
...
..
..
.
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.............
.....
..
..
.
..
................................................................
.....
...
..
..
..
..
..
...
...
....
......
................................
....................................
.....
...
...
..
...
..
....
..
..
..
.
..
.
.....
.......
..............
..........
..
.
.
.
.
.
.
.
.
.
.
.
.
with altogether n + m − 1 arrows, modulo the ideal generated by all
paths of length 2. Of course, if Λ is a finite-dimensional k-algebra
with radical square zero and Ext-quiver ∆(n, m2), and k is an
algebraically closed field, then Λ is Morita-equivalent to such an
algebra.
Also the following artin algebras with radical square zero and Ext-
quiver ∆(1, m2) may be of interest: the factor rings of the polynomial
ring Z[T1, . . . , Tm−1] modulo the square of the ideal generated by
some prime number p and the variables T1, . . . , Tm−1.
Jo
ur
na
l
A
lg
eb
ra
D
is
cr
et
e
M
at
h.
306 On radical square zero rings
References
[AR] M. Auslander, I. Reiten: On a generalized version of the Nakayama conjecture.
Proc. Amer. Math. Soc. 52 (1975), 69–74.
[ARS] M. Auslander, I. Reiten, S. Smalø: Representation Theory of Artin Algebras.
1995. Cambridge University Press.
[C] X.-W. Chen: Algebras with radical square zero are either self-injective or CM-free.
Proc. Amer. Math. Soc. Vol. 140, 93–98.
[JS] D. A. Jorgensen, L. M. Şega: Independence of the total reflexivity conditions
for modules. Algebras and Representation Theory 9 (2006), 217–226.
Contact information
C. M. Ringel Department of Mathematics, Shanghai Jiao
Tong University, Shanghai 200240, P. R. China
and King Abdulaziz University, P O Box 80200,
Jeddah, Saudi Arabia
E-Mail: ringel@math.uni-bielefeld.de
B.-L. Xiong Department of Mathematics, Beijing Univer-
sity of Chemical Technology, Beijing 100029,
P. R. China
E-Mail: xiongbaolin@gmail.com
Received by the editors: 24.05.2012
and in final form 17.01.2013.
|
| id | nasplib_isofts_kiev_ua-123456789-152245 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-24T11:44:42Z |
| publishDate | 2012 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Ringel, C.M. Xiong, B.-L. 2019-06-09T06:14:14Z 2019-06-09T06:14:14Z 2012 On radical square zero rings / C.M. Ringel, B.-L. Xiong // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 297–306. — Бібліогр.: 4 назв. — англ. 1726-3255 2010 MSC:16D90, 16G10; 16G70. https://nasplib.isofts.kiev.ua/handle/123456789/152245 Let Λ be a connected left artinian ring with radical square zero and with n simple modules. If Λ is not self-injective, then we show that any module M with Exti(M, Λ) = 0 for 1 ≤ i ≤ n + 1 is projective. We also determine the structure of the artin algebras with radical square zero and n simple modules which have a non-projective module M such that Exti(M, Λ) = 0 for 1 ≤ i ≤ n. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On radical square zero rings Article published earlier |
| spellingShingle | On radical square zero rings Ringel, C.M. Xiong, B.-L. |
| title | On radical square zero rings |
| title_full | On radical square zero rings |
| title_fullStr | On radical square zero rings |
| title_full_unstemmed | On radical square zero rings |
| title_short | On radical square zero rings |
| title_sort | on radical square zero rings |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152245 |
| work_keys_str_mv | AT ringelcm onradicalsquarezerorings AT xiongbl onradicalsquarezerorings |