A generalization of anti-commutative rings arising from 2-varieties

Let Rbe a nonassociative ring of characteristic not 2 or 3 which satisfies the identities (ab=ba) = (ac+ca)b, a(ac+ca) = b(ac+ca) and a²a = aa². We show that these rings are characterized as associative commutative rings with a type of biadditive mapping. From this characterization we show easily that simple rings are associative-commutative, or anti-commutative. Among the examples given, is a finite dimensional algebra which is solvable but not nilpotent.     

On Artinian rings satisfying the Engel condition

Let R be an Artinian ring, not necessarily with a unit, and let R ^o be the group of all invertible elements of R with respect to the operation a o b = a + b + ab. We prove that the group R ^o is a nilpotent group if and only if it is an Engel group and the quotient ring of the ring R by its Jacobson radical is commutative. In particular, R ^o is nilpotent if it is a weakly nilpotent group or an n-Engel group for some positive integer n. We also establish that the ring R is strictly Lie-nilpotent if and only if it is an Engel ring and the quotient ring of the ring R by its Jacobson radical is commutative. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1264–1270, September, 2006.     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

GOOD IDEALS IN MATRIX RINGS OVER COMMUTATIVE PIR's

Abstract

An ideal A of a ring R is called a good ideal if the coset product r ₁ r ₂ + A of any two cosets r ₁ + A and r ₂ + A of A in the factor ring R/A equals their set product (r ₁ + A) º (r ₂ + A): = {(r ₁ + a)(r ₂ + a ₂): a ₁, a ₂ ε A}. Good ideals were introduced in [3] to give a characterization of regular right duo rings. We characterize the good ideals of blocked triangular matrix rings over commutative principal ideal rings and show that the condition A º A = A is sufficient for A to be a good ideal in this class of matrix rings, none of which are right duo. It is not known whether good ideals in a base ring carries over to good ideals in complete matrix rings over the base ring. Our characterization shows that this phenomenon occurs indeed for complete matrix rings of certain sizes if the base ring is a blocked triangular matrix ring over a commutative principal ideal ring.     

Examples of Generic Lattice Ideals of Codimension 3

A commutative ring R is said to be strongly Hopfian if the chain of annihilators ann(a) ? ann(a ²) ? … stabilizes for each a ∈ R. In this article, we are interested in the class of strongly Hopfian rings and the transfer of this property from a commutative ring R to the ring of the power series R[[X]]. We provide an example of a strongly Hopfian ring R such that R[[X]] is not strongly Hopfian. We give some necessary and sufficient conditions for R[[X]] to be strongly Hopfian.     

When is a 2 × 2 Matrix Ring Over a Commutative Local Ring Quasipolar?

A ring R is quasipolar if for any a ∈ R, there exists p ² = p ∈ R such that p ∈ comm²(a), p + a ∈ U(R) and ap ∈ R ^qnil . In this article, we determine when a 2 × 2 matrix over a commutative local ring is quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be quasipolar. Consequently, we obtain several equivalent conditions for the 2 × 2 matrix ring over a commutative local ring to be quasipolar. Furthermore, it is shown that the 2 × 2 matrix ring over the ring of p-adic integers is quasipolar.     

Commutativity Conditions for Rings and Generalized 2-CN Rings

Firstly,the commutativity of rings is investigated in this paper.Let R be a ring with identity.Then we obtain the following commutativity conditions: (1) if for each x ∈ R\N(R) and each y ∈ R,(xy)k =xkyk for k =m,m + 1,n,n + 1,where m and n are relatively prime positive integers,then R is commutative;(2) if for each x ∈ R\J(R) and each y ∈ R,(xy)k =ykxk for k =m,m+ 1,m+2,where m is a positive integer,then R is commutative.Secondly,generalized 2-CN rings,a kind of ring being commutative to some extent,are investigated.Some relations between generalized 2-CN rings and other kinds of rings,such as reduced rings,regular rings,2-good rings,and weakly Abel rings,are presented.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

What lifts?

As is well known, idempotents in any ring R lift modulo any nil idea I. That is, if a ? R and (a₂-a) ? I there is an i ? I with (a + i)²-- (a + i) = 0. An idempotent is a zero of the polynomial x ²?x ², and a nil element satisfies x ⁿfor some n. Seen this way, lifting occurs in considerable generality We assume R has a unit, and handle the non-unital case at the end of this paper.     

ON SYMMETRIC UNITS IN GROUP ALGEBRAS

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K. The anti-automorphism g → g ^?1 of G can be extended linearly to an anti-automorphism a → a ^* of KG. Let S _* (KG) = {x ∈ U(KG) | x ^* = x} be the set of all symmetric units of U(KG). We consider the following question: for which groups G and commutative rings K it is true that S _* (KG) is a subgroup in U(KG). We answer this question when either a) G is torsion and K is a commutative G-favourable integral domain of characteristic p≥ 0 or b) G is non-torsion nilpotent group and KG is semiprime.     

Commutative rings with genus two annihilator graphs

Let R be a commutative ring and Z(R)* be its set of all nonzero zero-divisors. The annihilator graph of a commutative ring R is the simple undirected graph AG(R) with vertices Z(R)*, and two distinct vertices x and y are adjacent if and only if ann(xy)≠ann(x)∪ann(y). The notion of annihilator graph has been introduced and studied by Badawi [8 Badawi, A. (2014). On the annihilator graph of a commutative ring. Commun. Algebra 42(1):108–121.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. In this paper, we classify the finite commutative rings whose AG(R) are projective. Also we determine all isomorphism classes of finite commutative rings with identity whose AG(R) has genus two.     

Some criteria for the Cohen-Macaulay property and local cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

On divided commutative rings

Let R be a commutative ring with identity having total quotient ring T. A prime ideal P of R is called divided if P is comparable to every principal ideal of R. If every prime ideal of R is divided, then R is called a divided ring. If P is a nonprincipal divided prime, then P^-1 = { x ? T : xP ? P} is a ring. We show that if R is an atomic domain and divided, then the Krull dimension of R ≤ 1. Also, we show that if a finitely generated prime ideal containing a nonzerodivisor of a ring R is divided, then it is maximal and R is quasilocal.     

Distributive multiplication rings

A ringR is said to be a left (right)n-distributive multiplication ring, n>1 a positive integer, if aa₁a₂...a_n=aa₁aa₂...aa_n (a₁a₂...a_na=a₁aa₂a...a_na) for all a, a₁,...,a_n R. It will be shown that the semi-primitive left (right)n-distributive rings are precisely the generalized boolean ringsA satisfying aⁿ=a for all a A. An arbitrary left (right)n-distributive multiplication ring will be seen to be an extension of a nilpotent ringN satisfyingN ⁿ⁺¹=0 by a generalized boolean ring described above. Under certain circumstances it will be shown that this extension splits.     

The structure of symplectic groups over arbitrary commutative rings

LetR be an arbitrary commutative ring, andn be an integer ≥3. It is proved for any idealJ ofR thatESp _2n(R,J)=[ESp _2n(R),ESp _2n(J)]=[ESp _2n(R),ESp _2n(R,J)] =[ESp _2n(R),GSp _2n(R,J)]=[Sp _2n(R),ESp _2n(R,J)]. Furthermore, the problem of normal subgroups ofSp _2n(R) has an affirmative solution if and only ifaR=a ²R+2aR for eacha inR. This covers the relevant results of [4], [8], [10], [12] and [13]. Project Supported by the Science Fund of the Chinese Academy of Sciences     

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b? R with aR + bR = R, there exists y ? R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ? R), Theorem 6: A left or right N ₀o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N ₀-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N _o-continuous von Neumann regular ring is unit-regular     

Quasipolar Property of Generalized Matrix Rings

The article concerns the question of when a generalized matrix ring K _s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K _s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K _s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K _s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5 Cui , J. , Chen , J. ( 2011 ). When is a 2 × 2 matrix ring over a commutative local ring quasipolar? Comm. Alg. 39 : 3212 – 3221 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] are improved or extended.     

d-ideals,f d-ideals and prime ideals

Abstract

Let R be a commutative ring. An ideal I of R is called a d-ideal (f d-ideal) provided that for each a ∈ I (finite subset F of I) and b ∈ R, Ann(a) ? Ann(b) (Ann(F) ? Ann(b)) implies that b ∈ I. It is shown that, the class of z⁰-ideals (hence all sz⁰-ideals), maximal ideals in an Artinian or in a Kasch ring, annihilator ideals, and minimal prime ideals over a d-ideal are some distinguished classes of d-ideals. Furthermore, we introduce the class of f d-ideals as a subclass of d-ideals in a commutative ring R. In this regard, it is proved that the ring R is a classical ring with property (A) if and only if every maximal ideal of R is an f d-ideal. The necessary and sufficient condition for which every prime f d-ideal of a ring R being a maximal or a minimal prime ideal is given. Moreover, the rings for which their prime d-ideals are z⁰-ideals are characterized. Finally, we prove that every prime f d-ideal of a ring R is a minimal prime ideal if and only if for each a ∈ R there exists a finitely generated ideal , for some n ∈ ? such that Ann(a, I) = 0. As a consequence, every prime f d-ideal in a reduced ring R is a minimal prime ideal if and only if X= Min(R) is a compact space.     

Separative Ideals,Clean Elements,and Unit-Regularity

ABSTRACT

We prove that an ideal I of a regular ring R is separative if and only if each a ? R satisfying Rr(a)aR = Ra?(a)R = RaR(1 ? a)R ? I is unit-regular. If I is a separative ideal of a regular ring R, then each a ? R satisfying Rar(a²) = ?(a²)aR = R(a ? a²) R ? I is clean. Some applications are also obtained.