Involution rings with unique minimal *-biideal
The structure of certain involution rings which have exactly one minimal *-biideal is determined. In addition, involution rings with identity having a unique maximal biideal are characterized.
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| Cite this: | Involution rings with unique minimal *-biideal / D.I.C. Mendes // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 255-263. — Бібліогр.: 12 назв. — англ. |
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| citation_txt | Involution rings with unique minimal *-biideal / D.I.C. Mendes // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 255-263. — Бібліогр.: 12 назв. — англ. |
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| description | The structure of certain involution rings which have exactly one minimal *-biideal is determined. In addition, involution rings with identity having a unique maximal biideal are characterized.
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 2, pp. 255–263
© Journal “Algebra and Discrete Mathematics”
Involution rings with unique minimal *-biideal
D. I. C. Mendes∗
Communicated by V. M. Futorny
Abstract. The structure of certain involution rings which
have exactly one minimal *-biideal is determined. In addition, in-
volution rings with identity having a unique maximal biideal are
characterized.
1. Introduction
In the category of involution rings, it is not plausible to use the
concept of left (right) ideal, since a left (right) ideal which is closed under
involution is an ideal. An appropriate generalization which has been
efficient in playing the role of these in the case of involution rings is
that of *-biideal, first used by Loi [9] for proving structure theorems for
involution rings. For semiprime involution rings, Loi also investigated the
interrelation between the existence of minimal *-biideals and minimal
biideals and Aburawash [3] characterized minimal *-biideals by means
of idempotent elements. In [12], the author described minimal *-biideals
of an arbitrary involution ring. The structure and properties of certain
classes of right subdirectly irreducible rings (that is, rings in which the
intersection of all nonzero right ideals is nonzero) were determined by
∗Research supported by FEDER and Portuguese funds through the Centre for
Mathematics (University of Beira Interior) and the Portuguese Foundation for Sci-
ence and Technology (FCT- Fundação para a Ciência e a Tecnologia), Project PEst-
OE/MAT/UI0212/2013.
2010 MSC: Primary 16W10; Secondary 16D25, 16N20.
Key words and phrases: involution, biideal, nilpotent ring, local ring, subdirectly
irreducible ring, Jacobson radical.
256 Involution rings with unique minimal *-bi ideal
Desphande ([6] , [7]). It seems, therefore, pertinent to consider involution
rings in which the intersection of all nonzero *-biideals is nonzero. In
a broader setting, we shall determine the structure of involution rings,
belonging to certain classes, having exactly one minimal *-biideal.
All rings considered are associative and do not necessarily have identity.
Let us recall that an involution ring A is a ring with an additional unary
operation *, called involution, such that (a + b)∗ = a∗ + b∗, (ab)∗ = b∗a∗
and (a∗)∗ = a for all a, b ∈ A. An element of an involution ring A, which is
either symmetric or skew-symmetric, shall be called a *-element. A biideal
B of a ring A is a subring of A satisfying the inclusion BAB ⊆ B. An
ideal (biideal) B of an involution ring A is called a *-ideal (*-biideal) of A
if B is closed under involution; that is, B∗ = {a∗ ∈ A : a ∈ B} ⊆ B. An
involution ring A is semiprime if and only if, for any *-ideal I of A, I2 = 0
implies I = 0. An involution ring A is called *-subdirectly irreducible if
the intersection of all nonzero *-ideals of A (called the *-heart of A) is
nonzero.
2. Involution rings with unique minimal *-biideal
We begin by considering involution rings in which the intersection of
nonzero *-biideals is nonzero, which are obviously *-subdirectly irreducible.
These will be called *-bi-subdirectly irreducible rings. If p is a prime, then
Z(p) denotes the zero ring on the cyclic additive group of order p.
Proposition 1. Let A be a *-bi-subdirectly irreducible (with unique min-
imal *-biideal B). Then one of the following holds:
(i) A is a division ring with involution;
(ii) A ∼= D ⊕ Dop, where D is a division ring and D ⊕ Dop is endowed
with the exchange involution;
(iii) A is *-subdirectly irreducible involution ring with *-heart B ∼= Z(p)
for some prime p;
(iv) A is a *-subdirectly irreducible involution ring with *-heart H =
K ⊕ K∗, where K ∼= Z(2) ∼= K∗ and B = {a + a∗ : a ∈ K} ∼= Z(2).
Proof. Since the intersection of the nonzero *-biideals of A is nonzero, B
generates the *-heart H of A.
Case 1.
(
H2 6= 0
)
. Either H is a simple prime ring or H = K ⊕ K∗, where
the ideals K and K∗ of A are simple prime rings [5].
The *-biideal B is contained in every nonzero *-biideal B1 of H. Indeed,
0 6= B1HB1 is a *-biideal of A so that B ⊆ B1HB1 ⊆ B1. Therefore,
D. I. C. Mendes 257
H is a *-simple involution ring having a minimal *-biideal, namely B.
If H is simple prime, then H has a minimal left ideal L and L = He
for some idempotent element e in H [1]. Then 0 6= L∗L = e∗He is a
minimal *-biideal of H. So B = L∗L ⊆ L. The *-ideal H does not contain
other minimal left ideals besides L, for if L1 is a minimal left ideal of
H, then B = L∗
1L1 ⊆ L1. Now, 0 6= B ⊆ L ∩ L1 ⊆ L1 and since L
and L1 are minimal left ideals, it follows that L1 = L. Thus H = L
and H is a division ring. Since the *-essential *-ideal H has identity,
we have, by ([11] , Lemma 8) that A = H. Thus A is a division ring. If
H = K ⊕ K∗, then it is clear, from [1] , that K and K∗ have minimal left
ideals. Moreover, it can be deduced that K and K∗ have unique minimal
left ideals and this implies that K and K∗ are division rings. Consequently,
H = B and we have A = H = K ⊕ K∗ ∼= K ⊕ Kop endowed with the
exchange involution.
Case 2.
(
H2 = 0
)
. In this case, the *-biideal B ∼= Z (p) for some prime p,
according to ([12], Corollary 4(iii)). Moreover, every subgroup of H is a
biideal of A. By ([8], Proposition 6.2), H+, the additive group of H, is
an elementary abelian p-group and hence is a direct sum of cyclic groups
of order p. By our assumption on A, either H ∼= Z(p) or H = K ⊕ K∗,
where K ∼= Z(p) ∼= K∗. If p 6= 2, then the case H = K ⊕K∗ cannot occur,
for then {a + a∗ : a ∈ K} and {a − a∗ : a ∈ K} would be two distinct
minimal *-biideals of A.
The following corollary is immediate:
Corollary 2. An involution ring A is semiprime *-bi-subdirectly irre-
ducible if and only if it is one of the following types:
(i) a division ring;
(ii) D ⊕ Dop, where D is a division ring and D ⊕ Dop is endowed with
the exchange involution.
Next, we study certain classes of involution rings having exactly one
atom in their lattice of *-biideals. In the sequel, [a] and 〈a〉 denote,
respectively, the subring of A and the biideal of A generated by a ∈ A.
Furthermore, if B is a biideal of A with p elements (p prime), we let
AB =
{
a ∈ A : pa = 0 = a2 and a /∈ B
}
.
Lemma 3. Let A be a nilpotent involution p-ring (p prime). Then A has
a unique minimal *-biideal if and only if A is *-bi-subdirectly irreducible.
Proof. Let A have a unique minimal *-biideal B. Then B2 = 0, B contains
a minimal *-subring S of order p and B = S + SAS, the *-biideal
258 Involution rings with unique minimal *-bi ideal
generated by S. But SAS is a *-biideal of A and SAS = sAs for some
*-element s ∈ S. Hence, either sAs = 0 or sAs = B. The latter case
cannot occur, because then we would have 0 6= s = sas for some a ∈ A; a
contradiction with the fact that A is nilpotent. Therefore B = S ∼= Z (p).
Now we will show that S is contained in every nonzero *-biideal of A. Let
B1 be any nonzero *-biideal of A. There exists a nonzero *-element s1
in B1, of order p and such that s2
1 = 0. If s1As1 6= 0, then there exists a
nonzero *-element s2 in s1As1. Now s2As2 ⊆ s1As1 ⊆ B1. Continuing in
this way, we obtain a chain . . . ⊆ siAsi . . . ⊆ s2As2 ⊆ s1As1 ⊆ B1. Since
A is nilpotent, eventually we must obtain siAsi = 0 for some nonzero
*-element si ∈ B1. Hence 〈si〉 = [si] = S and so S ⊆ B1.
The converse is clear.
Proposition 4. If A is a nilpotent involution p-ring (p 6= 2 and p prime),
then the following conditions are equivalent:
(i) A has a unique minimal *-biideal B;
(ii) A is subdirectly irreducible with heart B ∼= Z (p) and, for each
a ∈ AB, at least one of the following holds: aAa 6= 0, aAa∗ 6= 0,
a∗Aa 6= 0, a∗a 6= 0, aa∗ 6= 0.
Proof. Suppose that (i) holds. From the Lemma 3, we know that B
is contained in every nonzero *-biideal of A. By Proposition 1, A is *-
subdirectly irreducible with *-heart B ∼= Z (p). Next, we show that A
is, in fact, subdirectly irreducible. Let I be any nonzero ideal of A such
that I 6= I∗. We claim that I ∩ I∗ 6= 0. Suppose, on the contrary, that
I ∩ I∗ = 0. Since A is nilpotent, there exists a least positive integer n > 2
such that In = 0. If n is even, let J = I
n
2 and if n is odd, let J = I
n+1
2 .
Hence J2 = JJ∗ = J∗J = 0. Then, for 0 6= j ∈ J such that pj = 0
and K = [j], it is easy to see that {k + k∗ : k ∈ K} and {k − k∗ : k ∈ K}
are two distinct *-biideals of A of order p, which is a contradiction with
our assumption. Therefore I ∩ I∗ 6= 0 and B ⊆ I ∩ I∗ ⊆ I. Hence A is
a subdirectly irreducible ring with heart B. Suppose that there exists
a ∈ AB such that aAa = aAa∗ = a∗Aa = 0 and a∗a = aa∗ = 0. If a is a *-
element, then [a] is a minimal *-biideal of A, which is a contradiction with
our assumption. If a is not a *-element, and T = [a], then {a+a∗ : a ∈ T}
and {a − a∗ : a ∈ T} are distinct minimal *-biideals of A, which is again
a contradiction.
Suppose that (ii) holds and let C be a minimal *-biideal of A and C
6= B. Clearly there exists a *-element a ∈ C ∩ AB and CAC = 0, whence
aAa = aAa∗ = a∗Aa = 0 and a∗a = aa∗ = 0, contradicting (ii).
D. I. C. Mendes 259
Corollary 5. If A is an involution p-ring (p 6= 2 and p prime) and
A2 = 0, then the following conditions are equivalent:
(i) A has a unique minimal *-biideal B;
(ii) A has a unique minimal *-subring B;
(iii) A has a unique minimal subring B;
(iv) A is subdirectly irreducible with heart B ∼= Z (p) and AB = ∅.
The following example illustrates that Corollary 5 is not true, in
general, when p = 2.
Example 6. The 2-ring A = Z (2)⊕Z (2), with the exchange involution, is
such that A2 = 0 and has a unique minimal *-biideal, B = {(0, 0) , (1, 1)}.
However, A is not subdirectly irreducible.
As usual, a ring A with identity 1 is called a local ring if A/J (A) is
a division ring, where J (A) denotes the Jacobson radical of A.
Proposition 7. Let A be a local involution ring of characteristic pn
(p 6= 2, p prime and n > 1) and with nonzero nilpotent Jacobson radical
J (A). Then
(i) if J (A) has a unique minimal *-biideal B, then B is the unique
minimal *-biideal of A;
(ii) B = {a ∈ A : aJ (A) = a∗J (A) = 0};
(iii) for a fixed nonzero b ∈ B, J (A) = {a ∈ A : ba = ba∗ = 0} =
{a ∈ A : aB = a∗B = 0};
(iv) for any b ∈ B, a ∈ J (A) \B, there exist a1, a2 ∈ J (A) \B such
that either b = aa1 = a2a (if a is a *-element) or b = (a + a∗) a1 =
a2 (a + a∗) (if a is not a *-element).
Proof. (i) Taking into account Proposition 1 and the fact that a local ring
contains only the trivial idempotents, it is clear that any minimal *-biideal
of A must be contained in the Jacobson radical J (A) of A. If J (A) has a
unique minimal *-biideal B, then we know that B ∼= Z(p) (Proposition 4).
Clearly, BAB ⊆ J (A) and so, if BAB 6= 0, then B ⊆ BAB. However,
this is impossible since J (A) is nilpotent. Thus BAB = 0 and so B is a
biideal of A. Since any minimal *-biideal C of A is contained in J (A),
we must have C = B.
(ii) From Proposition 1, B ∼= Z (p) and B is a *-ideal of A. Hence, for
any nonzero b ∈ B, bJ (A) ⊆ B implies that bJ (A) = 0 or bJ (A) = B.
However, the latter case cannot occur since J (A) is nilpotent. Similarly,
b∗J (A) = 0. Thus B ⊆ {a ∈ A : aJ (A) = a∗J (A) = 0}. Now to prove
the other inclusion, let a ∈ A such that aJ (A) = a∗J (A) = 0. Then
260 Involution rings with unique minimal *-bi ideal
a ∈ J (A) and a2 = 0. Moreover, we claim that pa = 0. Indeed, since
(p1)n = pn1 = 0, p1 is not invertible and hence p1 ∈ J (A) and pa =
a(p1) = 0. Taking into account Proposition 4, it follows that a ∈ B.
(iii) Let b be a fixed nonzero element in B. If x ∈ J (A), then
also x∗ ∈ J (A) and it follows from (ii) that bx = bx∗ = 0 and so
x ∈ {a ∈ A : ba = ba∗ = 0}. On the other hand, if x ∈ A such that bx =
bx∗ = 0, then x ∈ J (A), since J (A) contains all the zero divisors of A.
Since Ab = B, it is now clear that J (A) = {a ∈ A : ba = ba∗ = 0} =
{a ∈ A : Ba = Ba∗ = 0} = {a ∈ A : aB = a∗B = 0}.
(iv) Let b ∈ B and a ∈ J (A) \B. If a is a *-element, then b ∈
Aa ∩ aA. If, on the other hand, a is not a *-element, then b ∈ A (a + a∗) ∩
(a + a∗) A.
Lemma 8. Let A be a direct sum of rings, A = A1 ⊕ A2 ⊕ . . . ⊕ An, and
let B be a biideal of A. There exist biideals Bk of Ak, k = 1, 2, . . . , n, such
that B ⊆ B1 ⊕ B2 ⊕ . . . ⊕ Bn. In particular, if B is a minimal biideal of
A, then there exist minimal biideals Bk of Ak, k = 1, 2, . . . , n, such that
B ⊆ B1 ⊕ B2 ⊕ . . . ⊕ Bn.
Proof. For each k = 1, 2, . . . , n, consider the epimorphism πk : A1 ⊕ A2 ⊕
. . . ⊕ An → Ak given by πk ((a1, a2, . . . , an)) = ak and let πk (B) = Bk.
Then Bk is a biideal of Ak. For b = (b1, b2, . . . , bn) ∈ B, πk (b) = bk and
hence b ∈ B1 ⊕ B2 ⊕ . . . ⊕ Bn. Therefore B ⊆ B1 ⊕ B2 ⊕ . . . ⊕ Bn. Clearly,
if B is a minimal biideal of A, then πk (B) = Bk is a minimal biideal of
Ak, k = 1, 2, . . . , n.
For any prime p, let Ap denote, as usual, the p-component of an
involution ring A. In addition, an involution ring A is said to be a CI-
involution ring if every idempotent in A is central. Now we are in a
position to give the following classification theorem.
Theorem 9. Let A be a CI-involution ring with descending chain condi-
tion on *-biideals. Then A is *-bi-subdirectly irreducible if and only if A
is one of the following rings:
(i) A is a division ring with involution;
(ii) A ∼= D ⊕ Dop, where D is a division ring and D ⊕ Dop is endowed
with the exchange involution;
(iii) A is a local involution ring of characteristic pn (p prime and n > 1)
with nonzero nilpotent Jacobson radical, having a unique minimal
*-biideal;
D. I. C. Mendes 261
(iv) A ∼= L ⊕ Lop where each of the rings L and Lop is a local ring of
characteristic 2n (n > 1) with nonzero nilpotent Jacobson radical
having a unique minimal biideal and L ⊕ Lop is endowed with the
exchange involution;
(v) A is a nilpotent involution p-ring (p prime) having a unique minimal
*-biideal.
Proof. First we prove the direct implication. It is well-known that an
involution ring A has d.c.c. on *-biideals if and only if it is an artinian ring
with artinian Jacobson radical J (A) and J (A) is nilpotent. Moreover,
A = F ⊕ T , where the *-ideal T is the maximal torsion ideal of A and F
is a torsion-free *-ideal with identity and J (A) ⊆ T ([2] , [4] , [10]). Our
assumption on A implies that the intersection of all nonzero *-biideals
of A is a nonzero *-biideal and either A = T = Ap, for some prime p,
or A = F . Suppose that A = Ap. Since A is artinian, either Ap has a
nonzero idempotent or Ap is nilpotent. First, we consider the case when
Ap has a nonzero idempotent. Then Ap has a nonzero idempotent e which
is a *-element. Then e must be the identity of Ap. Indeed, if e is not the
identity of Ap, then eAp and (1 − e) Ap = {a − ea : a ∈ Ap} are nonzero
*-biideals with zero intersection, contradicting our assumption. If e is
the only nonzero idempotent in Ap, then, Ap, being artinian without
nontrivial idempotents, is a local ring of characteristic pn, for some integer
n > 1, having a unique minimal *-biideal, and so (i) or (iii) holds.
If there is another nonzero idempotent element f 6= e in Ap, then
f is not a *-element and ff∗ = 0. Indeed, if ff∗ 6= 0, then ff∗ = 1
and so f = ff∗, which is a contradiction with the fact that f is not a
*-element. Likewise, f∗f = 0. Hence f + f∗ is the identity element of Ap.
Furthermore, Ap = fAp ⊕ f∗Ap, where f and f∗ are the only nonzero
idempotents in fAp and f∗Ap, respectively. Hence each of the ideals
fAp and f∗Ap is a local ring of characteristic pn (n > 1) with nilpotent
Jacobson radical, having a unique minimal biideal. Thus (ii) or (iv) holds.
Notice that if p 6= 2 and S is the unique minimal biideal of fAp, then
{a + a∗ : a ∈ S} and {a − a∗ : a ∈ S} are two distinct minimal *-biideals
of Ap. If Ap is nilpotent, then (v) holds. Suppose now that A = F . From
Proposition 1 and the fact that A is torsion-free, it follows that A is
either a division ring of characteristic zero or A ∼= D ⊕ Dop, where D is
a division ring of characteristic zero and D ⊕ Dop is endowed with the
exchange involution.
Conversely, it is clear that the involution rings in (i) and (ii) are
*-bi-subdirectily irreducible (see [12]), and so are the involution rings
262 Involution rings with unique minimal *-bi ideal
in (iii) and (v). Taking into consideration Lemma 8, the involution rings
in (iv) have a unique minimal *-biideal, so the descending chain condition
on *biideals implies that these are *-bi-subdirectly irreducible.
3. Involution rings with unique maximal biideal
The next proposition states that an involution ring with identity which
has a unique maximal biideal B is a local involution ring with Jacobson
radical B. The proof is an easy adaptation of the well-known result that
if a ring A with identity has a unique maximal right ideal R, then R is in
fact an ideal of A and R = J (A).
Proposition 10. Let A be an involution ring with identity. If A has a
unique maximal biideal B, then B is a *-ideal of A and B = J (A).
Proof. Let a ∈ A. Then Ba is a biideal of A. If Ba 6= A, then Ba is
contained in a maximal biideal of A. Indeed, it is easily deduced, using
Zorn’s Lemma, that every biideal is contained in a maximal biideal.
Since B is the unique maximal biideal of A, Ba ⊆ B. On the other
hand, if Ba = A, then ba = 1 and b′a = a for certain b, b′ ∈ B. Now
0 6= ab = b′ab ∈ B; hence ab 6= 1 and 1 − b′ is not invertible and so
A(1 − b′) 6= A. But then A(1 − b′) is contained in a maximal biideal; that
is, 1 − b′ ∈ A(1 − b′) ⊆ B, whence 1 ∈ B, which is a contradiction. Thus
Ba = A is impossible and so B is a right ideal of A. Since every right
ideal is a biideal, we have that B is the unique maximal right ideal of
A. As is well-known, B is therefore an ideal of A, it is also the unique
maximal left ideal of A and B = J (A) is a *-ideal of A.
Corollary 11. A ring A with identity has a unique maximal biideal B
if and only if it has a unique maximal right (left) ideal.
Proof. The direct implication was proved in the previous proposition.
Conversely, let A have a unique maximal right ideal R and let B1 be a
maximal biideal of A. Then B1 ⊆ B1A ⊆ R and, since a right ideal is
also a biideal, the maximality of B1 implies that B1 = R.
We now terminate with a result which permits us to conclude that an
involution ring with identity having a unique maximal *-biideal may not
be a local ring.
Proposition 12. If B is a maximal *-biideal of an involution ring A
with identity, then one of the following conditions holds:
D. I. C. Mendes 263
(i) B is a maximal biideal of A;
(ii) there exist maximal biideals K and K∗ of A such that B = K ∩ K∗.
Proof. Let B be a maximal *-biideal of A. If B is not a maximal biideal of
A, then B is contained in a maximal biideal K of A. Since B is closed under
involution, B is also contained in K∗. Now B ⊆ K ∩ K∗, where K ∩ K∗ is
a *-biideal of A. The maximality of B now implies that B = K ∩ K∗.
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Contact information
D. I. C. Mendes Department of Mathematics, University of Beira
Interior, Covilhã, Portugal
E-Mail(s): imendes@ubi.pt
Received by the editors: 31.10.2012
and in final form 04.07.2015.
|
| id | nasplib_isofts_kiev_ua-123456789-155242 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T13:18:52Z |
| publishDate | 2016 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Mendes, D.I.C. 2019-06-16T14:34:50Z 2019-06-16T14:34:50Z 2016 Involution rings with unique minimal *-biideal / D.I.C. Mendes // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 255-263. — Бібліогр.: 12 назв. — англ. 1726-3255 2010 MSC:Primary 16W10; Secondary 16D25, 16N20. https://nasplib.isofts.kiev.ua/handle/123456789/155242 The structure of certain involution rings which have exactly one minimal *-biideal is determined. In addition, involution rings with identity having a unique maximal biideal are characterized. Research supported by FEDER and Portuguese funds through the Centre for Mathematics (University of Beira Interior) and the Portuguese Foundation for Science and Technology (FCT- Fundação para a Ciência e a Tecnologia), Project PEst-OE/MAT/UI0212/2013. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Involution rings with unique minimal *-biideal Article published earlier |
| spellingShingle | Involution rings with unique minimal *-biideal Mendes, D.I.C. |
| title | Involution rings with unique minimal *-biideal |
| title_full | Involution rings with unique minimal *-biideal |
| title_fullStr | Involution rings with unique minimal *-biideal |
| title_full_unstemmed | Involution rings with unique minimal *-biideal |
| title_short | Involution rings with unique minimal *-biideal |
| title_sort | involution rings with unique minimal *-biideal |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155242 |
| work_keys_str_mv | AT mendesdic involutionringswithuniqueminimalbiideal |