Abelian Rings Research Articles

Extending Abelian Rings: A Generalized Approach

We introduce a novel framework for assessing the centrality of idempotents within a ring by presenting a general concept that assigns a degree of centrality. This approach aligns with the previously established notions of semicentral and q-central idempotents by Birkenmeier and Lam. Specifically, we define an idempotent $e$ in a ring $R$ to be $n$-central, where $n$ is a positive integer, if $[e, R]^ne=0$, where $[x,y]$ represents the additive commutator $xy-yx$. If every idempotent in a ring $R$ is $n$-central, we refer to $R$ as $n$-Abelian. Our study lays the groundwork by presenting foundational results that support this concept and examines key features of $n$-central idempotents essential for appropriately categorizing $n$-Abelian rings among various generalizations of Abelian rings introduced in prior literature. We provide examples of $n$-central idempotents that do not fall under the categories of semicentral or $q$-central. Furthermore, we demonstrate that the ring of upper matrices $\mathbb{T}_n(R)$, where $R$ is Abelian, is an $n$-abelian. We also prove that a ring where all of its idempotents are $n$-central is an exchange ring if and only if the ring is clean.

On some generalizations of abelian rings

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings [Formula: see text] are always “J-abelian” in the sense that each idempotent of [Formula: see text] maps onto a central idempotent in [Formula: see text]. On the other hand, J-abelian rings turn out to be precisely the “strongly perspective rings” as well as the “strongly IC rings”, and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking [Formula: see text]-th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings.

Some new characterizations of (b,c)-inverses and Bott-Duffin (e,f)-inverses

The (b,c)-inverse and the Bott-Duffin (e,f)-inverse are two classes of outer inverses, a few characterizations of which have been presented by certain researchers. In this paper, we give some new characterizations of (b, c)-inverses and Bott-Duffin (e,f)-inverses. First, we present a number of ring theoretic characterizations of (b,c)-inverses. Then we characterize (b, c)-inverses by equations. Finally, we present some characterizations of Bott-Duffin (e,f)-inverses. More specifically, we use Bott-Duffin (e,f)- inverses to characterize some classes of rings, such as directly finite rings, Abelian rings and left min-abel rings.

An introduction to q-central idempotents and q-abelian rings

A ring idempotent is said to be quarter-central (or q-central for short) if . If all idempotents in a ring are q-central, is said to be a quarter-abelian ring (or more simply, a q-abelian ring). In this paper, we show that such a ring is characterized by the property that for all idempotents , where denotes the additive commutator . For any nonzero ring and any integer , we show that the ring of upper triangular matrices over is not q-abelian. On the other hand, is q-abelian iff is abelian (in the classical sense that all idempotents are central in ). A final section of this paper relates q-central idempotents to the notion of exchange rings and the study of regular, unit-regular, and strongly regular elements in arbitrary rings. From the viewpoint of currently prevailing generalizations of abelian rings in the literature, q-abelian rings are situated between the class of semiabelian rings and the class of strongly IC rings.

Donovan\u2019s conjecture, blocks with abelian defect groups and discrete valuation rings

We give a reduction to quasisimple groups for Donovan’s conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring {mathcal {O}}. Consequences are that Donovan’s conjecture holds for {mathcal {O}}-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan’s conjecture for {mathcal {O}}-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan’s conjecture for {mathcal {O}}-blocks is a consequence of conjectures predicting bounds on the {mathcal {O}}-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field.

On a property of polynomial rings over reversible rings

In this article, we first observe a sort of property of ideals generated by zero-dividing polynomials over reversible rings, in relation with products of ideals at zero. As a natural consequence of this result, we introduce the concept of a partially reflexive ring that is related to the reflexive ring property. It is shown that abelian π-regular rings are partially reflexive when Jacobson radicals are nilpotent. A ring R, in which the Jacobson radical J(R) is nilpotent and is simple, is shown to be partially reflexive; and it is proved that the polynomial ring over R is also partially reflexive. The structures of several kinds of algebraic systems are investigated with respect to the partial reflexivity.

Rings whose nilpotents form a multiplicative set

ABSTRACTNilpotents in Armendariz and abelian π-regular rings are multiplicatively closed. However, this fact need not hold in many kinds of rings. This article concerns a class of rings whose nilpotents are closed under multiplication. Rings contained in this class are said to be nilpotent-closed, and the structure of nilpotent-closed rings is investigated in relation with various situations which happen ordinarily in the study of noncommutative ring theory. In the procedure, various sorts of rings are investigated, so that they are nilpotent-closed. We also study familiar conditions under which nilpotents in nilpotent-closed rings form a subring.

English

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ℕ. We prove that M n (R) is nil clean if and only if R/J(R) is Boolean and M n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ℤ3, B or ℤ3 ⊕ B where B is a Boolean ring, and that M n (R) is weakly nil clean if and only if M n (R) is nil clean for all n ≥ 2.

Structure of Abelian rings

Let R be a ring with identity. We use J(R); G(R); and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R; respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ⊆ J(R) of R; that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2; 3; 4; and 5 orbits under the left regular action on X(R) by G(R): For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R); then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.

Strongly clean triangular matrices over abelian rings

We investigate the problem of determining when a triangular matrix ring over a strongly clean ring is, itself, strongly clean. We prove that, if R is a commutative clean ring, then Tn(R) is strongly clean for every positive n. In the more general case that R is an abelian clean ring, we provide sufficient conditions which imply that Tn(R) is strongly clean. We end with a brief consideration of the non-abelian case.

STRUCTURE OF IDEMPOTENTS IN RINGS WITHOUT IDENTITY

We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring proper- ties pass to power series rings and polynomial rings. It is also shown that �-regular rings are strongly �-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommuta- tive rings appropriate for the situations occurred naturally in studying standard ring theoretic properties. 1. Definitions and notations Throughout this paper, R denotes an associative ring without identity, unless otherwise stated. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). We let eij denote the usual matrix units with 1 in the (i,j)-position and zeros elsewhere, if the base ring has identity 1. Denote {(aij) ∈ Un(R) | the diagonal entries of (aij) are all equal} by Dn(R). Zn denotes the ring of integers modulo n. GF(p n ) denotes the Galois field of order p n . J(R) denotes the Jacobson radical of R. | | denotes the cardinality. The characteristic of R is denoted by charR, and |a| denotes the order of a ∈ R in the additive subgroup of R generated by a. R + means the additive Abelian group (R,+). The polynomial ring with an indeterminate x over R is denoted by R(x). While speaking about minimal ring in a certain class of rings, we refer to a ring with minimal order for rings in that class, due to Xue (21). The notation (S) stands for the two-sided ideal of R generated by ∅ 6 S ⊆ R, and we also write (a1,...,an) in place of (S) for simplicity when S = {a1,...,an}. A ring is called Abelian if every idempotent is central. The zero in a nil ring is the only idempotent and so every nil ring is Abelian. The class of Abelian

Central semicommutative rings

A ring R is central semicommutative if ab = 0 implies that aRb ⊆ Z(R) for any a, b ∈ R. Since every semicommutative ring is central semicommutative, we study sufficient condition for central semicommutative rings to be semicommutative. We prove that some results of semicommutative rings can be extended to central semicommutative rings for this general settings, in particular, it is shown that every central semicommutative ring is nil-semicommutative. We show that the class of central semicommutative rings lies strictly between classes of semicommutative rings and abelian rings. For an Armendariz ring R, we prove that R is central semicommutative if and only if the polynomial ring R[x] is central semicommutative. Moreover, for a central semicommutative ring R, it is proven that (1) R is strongly regular if and only if R is a left GP-V′-ring whose maximal essential left ideals are GW-ideals if and only if R is a left GP-V′-ring whose maximal essential right ideals are GW-ideals. (2) If R is a left SF and central semicommutative ring, then R is a strongly regular ring.

Small Modules over Abelian Regular Rings

We study the structure of infinitely generated small modules over abelian regular rings, i.e., modules over which the covariant functor Hom commutes with direct sums. It is shown that every infinitely generated small module has either an infinitely generated factor which is at most 22ω -generated or a countably generated essential submodule. As a consequence, we prove a module-theoretic criterion of steadiness for abelian regular rings.

Clean Index of Rings

A ring is called clean (resp., uniquely clean) if each of its elements can be (resp., uniquely) expressed as the sum of an idempotent and a unit. Motivated by recent work on uniquely clean rings in [6], we introduce the clean index of a ring R. For a ∈ R, let ℰ(a) = {e ∈ R: e 2 = e, a − e ∈ U(R)} where U(R) is the group of units of R and the clean index of R, denoted in(R), is defined by in(R) = sup{|ℰ(a)|: a ∈ R}. Thus, R is uniquely clean if and only if R is clean with in(R) = 1. So far, uniquely clean rings are the only clean rings whose structure is fully understood (see [6]). In this article, we characterize the (arbitrary) rings of clean indices 1, 2, 3 and determine the abelian rings of finite clean index. Applications to semipotent rings, semiprime rings, and clean rings are discussed.

ON A GENERALIZATION OF THE MCCOY CONDITION

We in this note consider a new concept, so called <TEX>$\pi$</TEX>-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of <TEX>$\pi$</TEX>-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of <TEX>$\pi$</TEX>-McCoy rings, observing the relations among <TEX>$\pi$</TEX>-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and (<TEX>$\pi-$</TEX>)regular rings. It is proved that the n by n full matrix rings (<TEX>$n\geq2$</TEX>) over reduced rings are not <TEX>$\pi$</TEX>-McCoy, finding <TEX>$\pi$</TEX>-McCoy matrix rings over non-reduced rings. It is shown that the <TEX>$\pi$</TEX>-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of <TEX>$\pi$</TEX>-McCoy rings are also examined.

On Abelian Rings

Let \alpha be an endomorphism of an arbitrary ring R with identity. In this note, we introduce the notion of \alpha-abelian rings which generalizes abelian rings. We prove that \alpha-reduced rings, \alpha-symmetric rings, \alpha-semicommutative rings and \alpha-Armendariz rings are \alpha-abelian. For a right principally projective ring R, we also prove that R is \alpha-reduced if and only if R is \alpha-symmetric if and only if R is \alpha-semicommutative if and only if R is \alpha-Armendariz if and only if R is \alpha-Armendariz of power series type if and only if R is \alpha-abelian.

Clean elements in abelian rings

Let R be a ring with identity. An element in R is said to be clean if it is the sum of a unit and an idempotent. R is said to be clean if all of its elements are clean. If every idempotent in R is central, then R is said to be abelian. In this paper we obtain some conditions equivalent to being clean in an abelian ring.

Further Results on K0-Groups with Ordered Structure

This paper is a continuation of our previous work [10]. By GAERS-1, we denote the class of generalized abelian exchange rings with stable range 1. In this paper, we first prove that for any ring R ∈ GAERS-1 and any ideal I of R, K0(R/I) is an archimedean ℓ-group, which is a natural generalization of [10, Theorem 5.3]. As applications, we establish explicit characterizations for the K0-simplicity of such rings in the sense of [3], and investigate the norm completeness of their K0-groups. Finally, we characterize the primitive idempotents in R by K0(R) with ordered structure, from which we can further determine completely the structure of K0(R).

Boolean Algebras, Generalized Abelian Rings, and Grothendieck Groups

ABSTRACT A ring R is called generalized Abelian if for each idempotent e in R, eR and (1 − e)R have no isomorphic nonzero summands. The class of generalized Abelian rings properly contains the class of Abelian rings. We denote by GAERS − 1 the class of generalized Abelian exchange rings with stable range 1. In this article we prove, by introducing Boolean algebras, that for any R ∈ GAERS − 1, the Grothendieck group K 0(R) is always an Archimedean lattice-ordered group, and hence is torsion free and unperforated, which generalizes the corresponding results of Abelian exchange rings. Our main technical tool is the use of the ordered structure of K 0(R)+, which provides a new method in the study of Grothendieck groups.

Left orders in Abelian regular rings

Left orders in Abelian regular rings