On GP-V-rings and Characterizations of Strongly Regular Rings

The purpose of this paper is to characterize strongly regular rings via MERT rings and weakly one-sided ideals. Many important equivalent conditions on strongly regular rings are shown.

Generalized Regular Ring

Using module class C R=Mx∈M,xRT=0,T∈I , we introduced the concepts of C R finitely generated module, C R finitely presented module and C R regular ring. We also discussed the criterion for C R regular ring,and the relations between C R regular ring and C R FP injective module.

I-semi-π-regular Rings

Let R be a ring and I an ideal of R. A ring R is called I-semi-π--regular if R/I is π-regular and idempotents of R can be strongly lifted modulo I. Characterizations of I-semi-π-regular rings are given and relations between semi-π-regular rings and semiregular rings are explored.

On Strongly Regular Rings

In this paper we investigate strongly regular rings. In terms of W-ideals of rings some characterizations of strongly regular rings are given.

K-THEORETICALLY COMPLETE VON NEUMANN REGULAR RINGS

The authors introduce the concept of Ko-complete ring and study the properties of Ko-complete unit-regular rings.

The Zero-divisor Graphs of Abelian Regular Rings

We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.

ON STRONGLY REGULAR RINGS

?The multiplication semigroup of strongly regular ring R in the light of semigroup is researched,hence some properties of strongly regular rings are obtained. The non-division strongly regular ring R is anilpotent semisimple ring without identity element. It is neither the Artin ring nor the Noether ring. The setidempotents of ring R is an infinite set without the maximum and minimal conditions,it is a unions of someorder sets and hai a non-well-ordered order set at least.

Generalized Semi-π-Regular Rings

In this paper, the concept of right generalized semi-π-regular rings is defined. We prove that these rings are non-trival generalizations of both right GP-injective rings and semi- π-regular rings. Some properties of these rings are studied and some results about generalized semiregular rings and GP-injective rings are extended.

A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the existence and the expression of the general solution tothe system are derived. As applications, necessary and sufficient conditions are given for thesystem of matrix equations A_1X = C_1 and A_3X = C_3 to have a bisymmetric solution, the system ofmatrix equations A_1X = C_1 and A_3XB_3 = C_3 to have a perselfconjugate solution over R with aninvolution and char R≠2, respectively. The representations of such solutions are also presented.Moreover, some auxiliary results on other systems over R are obtained. The previous known results onsome systems of matrix equations are special cases of the new results.

On the Projective Radicals of Regular Rings

We study the properties of projective radicals of regular rings. It is shown that the projective radical of a regular ring is left-right symmetric and a regular ring modulo its projective radical has zero projective radical. Also, we obtain a relation between projective radicals of a finitely generated projective module over a regular ring and its endomorphism ring, from which we give formulas about projective radicals of matrix rings and corners of a regular ring, and some equivalent conditions for a regular ring with zero projective radicals are given.

Regularity of Rings Whose Every Simple Singular Left R-module is GP-injective

In this paper, some new relations between GP-V'-rings and regular rings under certain special conditions have been found. It is proved that R is left self-injective regular with Soc(RR) ≠ 0 if and only if R is a left GP-V'-ring containing an injective maximal left ideal and Soc(RR)(?)Soc(RR). Moreover, for an MELT ring R, it is shown that R is regular if and only if R is a left GP-injective left GP-V'-ring.

Homomorphism and Isomorphism of Regular Power Ring

The upgrade of all kinds of algebraic structures has been emphasized with the development of fuzzy mathematics.The concept of hypergroup was raised first by Prof.LI Hong_xing in [1]and HX ring was done in [2].In this paper ,some properties of power ring and quasi_quotient ring are further studied based on paper [3～6].Especially,several theorems of homomorphism and isomorphism of regular power ring are established.

On Ideals of Regular Rings

in this paper,we investigate ideals of regular rings and give several characterizations for an ideal to satisfy the comparability.In addition,it is shown that.if I is a minimal two-sided ideal of a regular ring R,then I satisfies the Comparability if and only if I is separative.Furthermore,we prove that.for ideals with stable range one.Roth's problem has an affirmative solution.These extend the corresponding results on unit-regularity and one-sided unit-regularity.

GROUP ACTIONS ON VON NEUMANN REGULAR RINGS

Let A be a ring with indentity, G a finite group of automorphisms of A. The main result of this paper is that A/AG is Galois if and only if it is Frobenius and the module AGA (or AAG)is faithful. Moreover if |G| is invertible the author improves [2, Theorem 8] and [3, Theorem 8].

Ideal-comparability over Regular Rings

We introduce the concept of ideal-comparability condition for regular rings. Let I be an ideal of a regular ring R. If R satisfies the I-comparability condition, then R is one-sided unit-regular if and only if so is R/I. Also, we show that a regular ring R satisfies the general comparability if and only if the following hold： （1） R/I satisfies the general comparability; （2） R satisfies the general I-comparability condition; （3） The natural map B（R） → B（R/I） is surjective.

NOETHERIAN GT-REGULAR RINGS ARE REGULAR

It is proved that for a left Nootherian z-graded ring A, if every finitely generated gradedA-module has finite projective dimension (i.e., A is gr-regular) then every finitely generatedA-module has finite projective dimension (i.e., A is regular). Some applications of this resultto filtered rings and some classical cases are also given.

ON THE ORDER OF REGULAR RINGS

I. INTRODUCTION A ring R is called a regular ring if a∈aRa, for each a∈R. A solution to axa=a is called an inner inverse of a ,denoted by a^-. A solution to xax=x is called an outer inverse of a. A solution to both axa=a and xax=x is called a reflexive inverse of a, and denoted by a^+. Inner and reflexive inverses of an element are not unique in general. The sets of all in-

Comparability for ideals of regular rings

In this paper we investigate necessary and sufficient conditions under which the ideals possess comparability structure. For regular rings, we prove that every square matrix over ideals satisfying general comparability admits a diagonal reduction by quasi invertible matrices.

When Exchange Rings are Von Neumann Regular

We study when exchange rings are von Neumann regular. An exchange ring R with primitive factors Artinian is von Neumann regular, if the Jacobson radical of any indecomposable homomorphic image of R is T-nilpotent, and if any indecomposable homomorphic image of R is semiprime. Every indecomposable semiprimitive factor ring of R is regular, if R is an exchange ring such that every left primitive factor ring of R is a ring of index at most n and if R has nil-property.

Generalizations of von Neumann regular rings, PP rings, and Baer rings

We introduce the notions of IDS modules, IP modules, and Baer＊ modules, which are new generalizations of von Neumann regular rings, PP rings, and Baer rings, respectively, in a general module theoretic setting. We obtain some characterizations and properties of IDS modules, IP modules and Baer modules. Some important classes of rings are characterized in terms of IDS modules, IP modules, and Baer modules.