Morphic group rings

An element a in a ring R is called left morphic if there exists b∈R such that l_R(a)=Rb and l_R(b)=Ra, where l_R(a) denotes the left annihilator of a in R. The ring R is called left morphic if every element of R is left morphic. Left morphic rings have been studied by Nicholson and Sánchez Campos. In this paper, the question of when a group ring is left morphic is discussed in great detail and various morphic group rings are identified.     

McCoy rings and zero-divisors

We investigate relations between the McCoy property and other standard ring theoretic properties. For example, we prove that the McCoy property does not pass to power series rings. We also classify how the McCoy property behaves under direct products and direct sums. We prove that McCoy rings with 1 are Dedekind finite, but not necessarily Abelian. In the other direction, we prove that duo rings, and many semi-commutative rings, are McCoy. Degree variations are defined, studied, and classified. The McCoy property is shown to behave poorly with respect to Morita equivalence and (infinite) matrix constructions.     

On the Behrens radical of matrix rings and polynomial rings

It is shown that the Behrens radical of a polynomial ring, in either commuting or non-commuting indeterminates, has the form of “polynomials over an ideal”. Moreover, in the case of non-commuting indeterminates, for a given coefficient ring, the ideal does not depend on the cardinality of the set of indeterminates. However, in contrast to the Brown-McCoy radical, it can happen that the polynomial ring R[X] in an infinite set X of commuting indeterminates over a ring R is Behrens radical while the polynomial ring R〈X〉 in an infinite set Y of non-commuting indeterminates over R is not Behrens radical. This is connected with the fact that the matrix rings over Behrens radical rings need not be Behrens radical. The class of Behrens radical rings, which is closed under taking matrix rings, is described.     

Some families of strongly clean rings

A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute with each other. For a commutative local ring R and for an arbitrary integer n?2, the paper deals with the question whether the strongly clean property of M_n(R[[x]]), , and M_n(RC₂) follows from the strongly clean property of M_n(R). This is ‘Yes’ if n=2 by a known result.     

Rings in which elements are uniquely the sum of an idempotent and a unit that commute

A ring is called uniquely clean if every element is uniquely the sum of an idempotent and a unit. The rings described by the title include uniquely clean rings, and they arise as triangular matrix rings over commutative uniquely clean rings. Various basic properties of these rings are proved and many examples are given.     

Strong lifting splits

The concept of an enabling ideal is introduced so that an ideal I is strongly lifting if and only if it is lifting and enabling. These ideals are studied and their properties are described. It is shown that a left duo ring is an exchange ring if and only if every ideal is enabling, that Zhou’s δ-ideal is always enabling, and that the right singular ideal is enabling if and only if it is contained in the Jacobson radical. The notion of a weakly enabling left ideal is defined, and it is shown that a ring is an exchange ring if and only if every left ideal is weakly enabling. Two related conditions, interesting in themselves, are investigated: the first gives a new characterization of δ-small left ideals, and the second characterizes weakly enabling left ideals. As an application (which motivated the paper), let M be an I-semiregular left module where I is an enabling ideal. It is shown that m∈M is I-semiregular if and only if m−q∈IM for some regular element q of M and, as a consequence, that if M is countably generated and IM is δ-small in M, then where for each i.     

A note on completeness and strongly clean rings

Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we investigate the classical problem of lifting idempotents, in order to consolidate and extend these results. Our main result is that if R$R$ is a ring which is complete with respect to an ideal I$I$ and if x$x$ is an element of R$R$ whose image in R/I$R/I$ is strongly π$\pi$-regular, then x$x$ is strongly clean in R$R$. This generalizes Theorem 2.1 of Chen and Zhou (2007)  [9].     

Duo group rings

It is shown that the group algebra of a torsion group G over a field K is duo if and only if it is reversible.     

Clean endomorphism rings

A ring is called clean if every element is the sum of an idempotent and a unit. It is shown that the endomorphism ring of a projective right module over a right perfect ring is clean.Received: 6 January 2003     

Reversible and symmetric rings

We determine the precise relationships among three ring-theoretic conditions: duo, reversible, and symmetric. The conditions are also studied for rings without unity, and the effects of adjunction of unity are considered.     

An ensemble of idempotent lifting hypotheses

Lifting idempotents modulo ideals is an important tool in studying the structure of rings. This paper lays out the consequences of lifting other properties modulo ideals, including lifting of von Neumann regular elements, lifting isomorphic idempotents, and lifting conjugate idempotents. Applications are given for IC rings, perspective rings, and Dedekind-finite rings, which improve multiple results in the literature. We give a new characterization of the class of exchange rings; they are rings where regular elements lift modulo all left ideals.We also uncover some hidden connections between these lifting properties. For instance, if regular elements lift modulo an ideal, then so do isomorphic idempotents. The converse is true when units lift. The logical relationships between these and several other important lifting properties are completely characterized. Along the way, multiple examples are developed that illustrate limitations to the theory.     

Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

For a commutative ring R with set of zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we show that Γ(T(R)) and Γ(R) are isomorphic as graphs, where T(R) is the total quotient ring of R, and that Γ(R) is uniquely complemented if and only if either T(R) is von Neumann regular or Γ(R) is a star graph. We also investigate which cardinal numbers can arise as orders of equivalence classes (related to annihilator conditions) in a von Neumann regular ring.     

Commutativity conditions for rings: 1950–2005

During the last 55 years there have been many results concerning conditions that force a ring to be commutative. These results were stimulated by Jacobson's famous result and were extensively developed by Herstein. This paper will survey the area by organizing the results according to whether they come from variations on Herstein's conditions, depend on general polynomial conditions, depend on the presence of a derivation, or whether a ring has special properties that make commutativity more easily accessible. Finally, the most recent conditions concern product sets and lead to results in a new area of inquiry.     

Weakly regular rings with ACC on annihilators and maximality of strongly prime ideals of weakly regular rings

Ramamurthi proved that weak regularity is equivalent to regularity and biregularity for left Artinian rings. We observe this result under a generalized condition. For a ring R satisfying the ACC on right annihilators, we actually prove that if R is left weakly regular then R is biregular, and that R is left weakly regular if and only if R is a direct sum of a finite number of simple rings. Next we study maximality of strongly prime ideals, showing that a reduced ring R is weakly regular if and only if R is left weakly regular if and only if R is left weakly π-regular if and only if every strongly prime ideal of R is maximal.     

On a characterization of complete matrix rings

Peter R. Fuchs established in 1991 a new characterization of complete matrix rings by showing that a ringR with identity is isomorphic to a matrix ringM _n (S) for some ringS (and somen 2) if and only if there are elementsx andy inR such thatx ^n–1 0,x ⁿ=0=y ²,x+y is invertible, and Ann(x ^n–1)Ry={0} (theintersection condition), and he showed that the intersection condition is superfluous in casen=2. We show that the intersection condition cannot be omitted from Fuchs' characterization ifn3; in fact, we show that if the intersection condition is omitted, then not only may it happen that we do not obtain a completen ×n matrix ring for then under consideration, but it may even happen that we do not obtain a completem ×m matrix ring for anym2.     

A simple proof of a theorem on quasi-Baer rings

We give a simple proof of a theorem, due to Birkenmeier, Kim and Park, which states that if $R[x, x^{-1}]$ or $R[[x, x^{-1}]]$ is a quasi-Baer ring then R is a quasi-Baer ring. Received: 8 April 2002     

Induced liftings, exchange rings and semi-perfect algebras

Several important classes of rings can be characterized in terms of liftings of idempotents with respect to various ideals: classical examples are semi-perfect rings, semi-regular rings and exchange rings. We begin with a study of some extensions of the concept of idempotent lifting and prove the generalizations of some classical lifting theorems. Then we describe the method of induced liftings, which allows us to transfer liftings from a ring to its subrings. Using this method we are able to show that under certain assumptions a subring of an exchange ring is also an exchange ring, and to prove that a finite algebra over a commutative local ring is semi-perfect, provided it can be suitably represented in an exchange ring.     

On strongly Π-regular ideals

In this paper, we introduce the concept of strongly π-regular ideal of a ring. We prove that every square regular matrix over a strongly π-regular ideal of a ring admits a diagonal reduction.