EXCHANGE MORITA RINGS

In this paper we characterize the largest exchange ideal of a ring R as the set of those elements x ∈ R such that the local ring of R at x is an exchange ring. We use this result to prove that if R and S are two rings for which there is a quasi-acceptable Morita context, then R is an exchange ring if and only if S is an exchange ring, extending an analogue result given previously by Ara and the second and third authors for idempotent rings. We introduce the notion of exchange associative pair and obtain some results connecting the exchange property and the possibility of lifting idempotents modulo left ideals. In particular we obtain that in any exchange ring, orthogonal von Neumann regular elements can be lifted modulo any one-sided ideal.     

A note on weakly clean rings

Let R be an associative ring with identity. An element x??R is said to be weakly clean if x=u+e or x=u?e for some unit u and idempotent e in R. The ring R is said to be weakly clean if all of its elements are weakly clean. In this paper we obtain an element-wise characterization of abelian weakly clean rings. A relation between unit regular rings and weakly clean rings is also obtained.     

Reflexive Property of Rings

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.     

Regularity and Related Rings

A ring R is a Garcia ring provided that the product of two regular elements is unit-regular. We prove that every regular element in a Garcia ring R is the sum/difference of an idempotent and a unit. Furthermore, we prove that every regular element in a weak Garcia ring is the sum of an idempotent and a one-sided unit. These extend several known theorems on (one-sided) unit-regular rings to wider classes of rings with sum summand property.     

Semiprime rings with bounded indices of nilpotent elements

This paper deals with the structure of semiprime rings for which the indices of the nilpotent elements are bounded. It is shown that the complete right ring of quotients of such a ring is a regular, right self-injective ring in which each finitely generated ideal is generated by a central idempotent. The indices of the nilpotent elements of the factor ring of such a ring with respect to a minimal prime ideal do not exceed the upper bound of the indices of the nilpotent elements of the original ring. A criterion for the regularity (in the sense of von Neumann) of such rings is obtained. Also investigated are right completely idempotent rings with bounded indices of nilpotent elements (it is shown, in particular, that each nonzero ideal of such a ring contains a nonzero central idempotent).Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 237–249, 1988.     

Rings with x(yz)=z(yx)

In non-associative rings the associative law is replaced by various weaker identities. In this paper the identity x.yz = z.yx is considered. Let R be a ring satisfying this identity. It is obvious that if R has an identity element, then R is both commutative and associative. It is shown that if R is prime, third power associative, 2-torsion free, and has a nonzero idempotent, then this idempotent must be an identity and hence R must be commutative and associative.     

On Algebras and Their Radicals

In this paper, the concept of Algebras over alternative (Lie, Jordan, respectively) rings is introduced. Some results on the quasi-direct sum of associative rings are extended and generalized. Also, associative Algebras are described by some concrete ring homomorphisms.     

Jordan isomorphisms of generalized structural matrix rings

We describe the sub-bimodules of matrix bimodules over two structural matrix rings. Structural matrix bimodules arise as particular such sub-bimodules, and we discuss when such a bimodule is faithful or indecomposable. As an application, we obtain a large class of rings whose Jordan isomorphisms are either ring isomorphisms or ring anti-isomorphisms. Complete upper block triangular matrix rings over 2-torsion-free indecomposable rings are elements of this class.     

Local rings of exchange rings

We characterize the exchange property for non-unital rings in terms of their local rings at elements,and we use this characterization to show that the exchange property is Morita invariant for idempotent rings.We also prove that every ring contains a greatest exchange idela(with respect to the inclusion).     

Radicals of Jordan rings connected with alternative rings

Subject to a certain restriction on the additive group of an alternative ring A, we prove that R(A)=R(A⁽⁺⁾), where A⁽⁺⁾ is a Jordan ring and R is one of the following radicals: the Jacobson radical, the upper nil-radical, the locally nilpotent radical, or the lower nil-radical. For the proof of these relationships Herstein's well-known construction for associative rings is generalized to alternative rings.     

Galois subrings of simple rings

Suppose R is a finite direct sum of simple associative rings and G is a finite group of auto-morphisms of the ring R. It is shown that if there is no additive ¦G¦-torsion in R, then the subring of elements of R that are fixed under G is a finite direct sum of simple rings.     

Classes of Almost Clean Rings

A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular module is almost clean and that every right CS (i.e. right extending) and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite AW ^*-algebras and noetherian Leavitt path algebras in particular, are almost clean. We say that a ring R is special clean (special almost clean) if each element a can be decomposed as the sum of a unit (regular element) u and an idempotent e with aR?∩?eR?=?0. The Camillo-Khurana Theorem characterizes unit-regular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasi-continuous and right nonsingular ring is left and right Rickart. If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unit-regular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasi-continuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean. Finally, we adapt some of our results to rings with involution: a *-ring is *-clean (almost *-clean) if each of its elements is the sum of a unit (regular element) and a projection (self-adjoint idempotent). A special (almost) *-clean ring is similarly defined by replacing “idempotent” with “projection” in the appropriate definition. We show that an abelian *-ring is a Rickart *-ring if and only if it is special almost *-clean, and that an abelian *-ring is *-regular if and only if it is special *-clean.     

Special jordan pairs

There are some important Jordan pairs contained in the free associative pair, e.g. the free special Jordan pair and the Jordan pair of all symmetric elements under the reversal involution. We study the connection between these two Jordan pairs in the case when the ring of scalars contains ½.     

Simple Jordan superalgebras with associative nil-semisimple even part

Under study are the simple infinite-dimensional abelian Jordan superalgebras not isomorphic to the superalgebra of a bilinear form. We prove that the even part of such superalgebra is a differentially simple associative commutative algebra, and the odd part is a finitely generated projective module of rank 1. We describe unital simple Jordan superalgebras with associative nil-semisimple even part possessing two even elements which induce a nonzero derivation.     

Monomorphisms in the category of firm modules

Abstract

In this article, we consider the category of unitary right modules over an (associative) ring and the category of firm right modules over an idempotent ring. We study monomorphisms in these categories and give conditions under which morphisms are monomorphisms in the category of firm modules. We also prove that the lattice of categorically defined subobjects of a firm module is isomorphic to the lattice of unitary submodules of that module.     

K-本原环的结构

研究K-本原环.证明了素环R是K-本原环当且仅当R含有一个非零理想I是K-本原环,当且仅当eRe是K-本原环,其中e是R的非零幂等元.并证明了GPI素环是K-本原环.推广了文献中的相应结果.     

On train algebras of rank 4

The existence of idempotent elements in train algebras of rank greater than 3 is an open question to be solved. Recent H. Guzzo results [7] on train algebras of rank 4 are based on the underlying assumption of the existence of an idempotent. In the present paper we establish the conditions that ensure the existence of such an idempotent. We also give additional properties on the Peirce decomposition which allow us to characterize some train algebras of rank 4. Finally, we give a characterization of the train algebras of rank 4 which are power-associative algebras or Jordan algebras.     

Derivations of a matrix ring containing a subring of triangular matrices

We describe the derivations (including Jordan and Lie ones) of finitary matrix rings, containing a subring of triangular matrices, over an arbitrary associative ring with identity element.     

Structure of some Unital Simple Jordan Superalgebras with Associative Even Part

Studying the unital simple Jordan superalgebras with associative even part, we describe the unital simple Jordan superalgebras such that every pair of even elements induces the zero derivation and every pair of two odd elements induces the zero derivation of the even part. We show that such a superalgebra is either a superalgebra of nondegenerate bilinear form over a field or a four-dimensional simple Jordan superalgebra.