Semiregular Rings Research Articles

On Semiregular Endomorphism Rings and the Dual of Yamagata’s Theorem

On Semiregular Endomorphism Rings and the Dual of Yamagata’s Theorem

On 1-semiregular and 2-semiregular rings

In this paper, we are mainly interested in the two questions “which are the commutative rings on which every finitely presented modules is [Formula: see text]-periodic (respectively, [Formula: see text]-periodic)?”. It is proved that these kinds of rings are particular cases of semiregular rings. So, we call them [Formula: see text]-semiregular and [Formula: see text]-semiregular rings, respectively. We establish characterizations of these rings in terms of various classical notions and we provide several examples of such rings.

A new characterization of commutative semiregular rings

A commutative ring R is called J-rad clean in case, for any r ∈ R, there is an idempotent e ∈ R such that r−e ∈ U(R) and re ∈ J(R), where U(R) and J(R) denote the set of units and the Jacobson radical of R, respectively. Also, a ring R is called semiregular if R/J(R) is regular in the sense of von Neumann and idempotents lift modulo J(R). We demonstrate that these two concepts are, actually, equivalent and we portray a portion of the properties of this class of rings. In particular, as a direct application, we prove that the commutative group ring RG is J-rad clean if, and only if, R is a commutative J-rad clean ring and G is a torsion abelian group, provided that J(R) is nil.

Trivial extensions subject to semi-regularity and semi-coherence

In this paper, we investigate the transfer of Matlis’ semi-regularity and semi-coherence in trivial ring extensions issued from rings (with zero-divisors). We use the obtained results to enrich the literature with new examples of semi-regular or semi-coherent rings issued as trivial extensions and validate some questions left open in the literature.

On rings with annihilator condition

In this paper we study rings R with the property that every finitely generated ideal of R consisting entirely of zero divisors has a nonzero annihilator. The class of commutative rings with this property is quite large; for example, noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose classical ring of quotients are von Neumann regular. We continue to study conditions under which right mininjective rings, right FP-injective rings, right weakly continuous rings, right extending rings, one sided duo rings, semiregular rings and semiperfect rings have this property.

I*0-Modules

We study rings over which every module is an I*0 -module dual to I0-module. We describe semiregular rings over which every module is at the same time I*0 -module and I0-module. We give a description of rings over which every module is a direct sum of injective module and SV -module. We investigate relations between weakly Baer modules and I*0 -modules.

PSEUDOPOLAR MATRIX RINGS OVER LOCAL RINGS

A ring R is called pseudopolar if for every a ∈ R there exists p2 = p ∈ R such that p ∈ comm 2(a), a + p ∈ U(R) and akp ∈ J(R) for some positive integer k. Pseudopolar rings are closely related to strongly π-regular rings, uniquely strongly clean rings, semiregular rings and strongly π-rad clean rings. In this paper, we completely characterize the local rings R for which M2(R) is pseudopolar.

On Quasipolar Rings

The notion of quasipolar elements of rings was introduced by Koliha and Patricio in 2002. In this paper, we introduce the notion of quasipolar rings and relate it to other familiar notions in ring theory. It is proved that both strongly π-regular rings and uniquely clean rings are quasipolar, and quasipolar rings are strongly clean, but no two of these classes of rings are equivalent. For commutative rings, quasipolar rings coincide with semiregular rings. It is also proved that every n × n upper triangular matrix ring over any commutative uniquely clean ring or commutative local ring is quasipolar.

Pseudo Drazin inverses in associative rings and Banach algebras

Motivated by strongly π-regular elements and quasipolar elements, we introduce the concept of pseudopolar elements. An element a∈R is called pseudopolar if there exists p∈R such that p2=p∈comm2(a),a+p∈U(R)andakp∈J(R) for some positive integer k. This concept can be used exactly to define a pseudo Drazin inverse in associative rings and Banach algebras. We connect pseudopolar rings with strongly π-regular rings, semiregular rings, uniquely strongly clean rings and uniquely bleached local rings. Some basic properties of pseudo Drazin inverses are obtained in associative rings and Banach algebras.

On lifting of idempotents and semiregular endomorphism rings

On lifting of idempotents and semiregular endomorphism rings

A Generalization of Semiregular and Almost Principally Injective Rings

In this article, we call a ring R right almost I-semiregular for an ideal I of R if for any a ∈ R, there exists a left R-module decomposition lRrR(a) = P ⊕ Q such that P ⊆ Ra and Q ∩ Ra ⊆ I, where l and r are the left and right annihilators, respectively. This generalizes the right almost principally injective rings defined by Page and Zhou, I-semiregular rings defined by Nicholson and Yousif, and right generalized semiregular rings defined by Xiao and Tong. We prove that R is I-semiregular if and only if for any a ∈ R, there exists a decomposition lRrR(a) = P ⊕ Q, where P = Re ⊆ Ra for some e2 = e ∈ R and Q ∩ Ra ⊆ I. Among the results for right almost I-semiregular rings, we show that if I is the left socle Soc (RR) or the right singular ideal Z(RR) or the ideal Z(RR) ∩ δ(RR), where δ(RR) is the intersection of essential maximal left ideals of R, then R being right almost I-semiregular implies that R is right almost J-semiregular for the Jacobson radical J of R. We show that δl(eRe) = e δ(RR)e for any idempotent e of R satisfying ReR = R and, for such an idempotent, R being right almost δ(RR)-semiregular implies that eRe is right almost δl(eRe)-semiregular.

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.

Rings with Multiplicative Sets of Primitive Idempotents

Let R be a ring and M(R) the set consisting of zero and primitive idempotents of R. We study the rings R for which M(R) is multiplicative. It is proved that if R has a complete finite set of primitive orthogonal idempotents, then R is a finite direct product of connected rings precisely when M(R) is multiplicative. We prove that if R is a (von Neumann) regular ring with M(R) multiplicative, then every primitive idempotent in R is central. It is also shown that this does not happen even in semihereditary and semiregular rings. Let R be an arbitrary ring with M(R) multiplicative and e ∈ R be a primitive idempotent, then for every unit u ∈ R, it is proved that eue is a unit in eRe. We also prove that if M(R) is multiplicative, then two primitive idempotents e and f in R are conjugates, i.e., f = ueu −1 for some u ∈ U(R), if and only if ef ≠ 0.

Generalizations of Semiregular Rings#

In this article, we call a ring R right generalized semiregular if for any a ∈ R there exist two left ideals P, L of R such that lr(a) = P⊕ L, where P ⊆ Ra and Ra ∩ L is small in R. The class of generalized semiregular rings contains all semiregular rings and all AP-injective rings. Some properties of these rings are studied and some results about semiregular rings and AP-injective rings are extended. In addition, we call a ring R semi-π-regular if for any a ∈ R there exist a positive integer n and e 2 = e ∈ a n R such that (1 − e)a n ∈ J(R), the Jacobson radical of R. It is shown that a ring R is semi-π-regular if and only if R/J(R) is π-regular and idempotents can be lifted modulo J(R).

Strong lifting

If I is an ideal of a ring R, we say that idempotents lift strongly modulo I if, whenever a 2 − a ∈ I , there exists e 2 = e ∈ a R (equivalently e 2 = e ∈ R a ) such that e − a ∈ I . The higher socles of R all enjoy this property, as does the Jacobson radical J if idempotents lift modulo J. Many of the useful, basic properties of lifting modulo J are shown to extend to any ideal I with strong lifting, and analogs of the semiperfect and semiregular rings are studied. A number of examples are given that limit possible extensions of the results.

Cotilting versus pure-injective modules

Let R and S be arbitrary associative rings. A left R-module R W is said to be cotilting if the class of modules cogenerated by R W coincides with the class of modules for which the functor Ext 1 R(-, W) vanishes. In this paper we characterize the cotilting modules which are pure-injective. The two notions seem to be strictly connected: Indeed all the examples of cotilting modules known in the literature are pure-injective. We observe that if R W S is a pure-injective cotilting bimodule, both R and S are semiregular rings and we give a characterization of the reflexive modules in terms of a suitable linear compactness notion.

Semiregular, Semiperfect and Perfect Rings Relative to an Ideal

Let I be an ideal of a ring R. Consider the following conditions on R: 1. If X is a finitely generated submodule of a finitely generated projective module P, then X = A ○+ B where A is a summand of P and B ⊆ P . I. 2. If X is a submodule of a finitely generated projective module P, then X = A ○+ B where A is a summand of P and B ⊆ P . I. 3. If X is a submodule of a projective module P, then X = A ○+ B where A is a summand of P and B ⊆ P . I. When I is the Jacobson radical J(R) of R, these conditions characterize semiregular rings, semiperfect rings and right perfect rings, respectively. In this paper we completely characterize these conditions for the cases when I is the right singular ideal, or the right socle, or the intersection of any two of the three ideals. As applications, structure theorems are obtained for right CEP-rings R with J(R) 2 = 0 and for QF-rings R with J(R) 2 = 0.

Semiregular, Weakly Regular, and Π-Regular Rings

Publisher Summary This chapter discusses the semi-regular, weakly regular and π-regular rings. All rings are associative and have nonzero identity elements. Every strongly π-regular ring is π-regular. The factor ring of the integers with respect to the ideal generated by the integer four is a strongly re-regular ring which is not a regular ring. For a ring A, the Jacobson radical of A is denoted by J (A) which is equal to the intersection of all maximal right ideals of A by definition. A ring A is said to be semi-regular if A / J (A) is a regular ring, and all idempotents of A / J (A) are images of idempotents of A for the natural epimorphism A → A / J (A). The classes of semi-regular and π-regular rings are quite large. Every regular ring is a semi-regular π-regular ring. All right or left Artinian rings are semi-regular π-regular rings, and the endomorphism ring of any injective module is a semi-regular ring.

Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P → M such that P is projective and Ker(p) is δ-small in P, then we say that P is a projective δ-cover of M. A ring R is called δ-perfect (resp., δ-semiperfect, δ-semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective δ-cover. The class of all δ-perfect (resp., δ-semiperfect, δ-semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of δ-perfect, δ-semiperfect, and δ-semiregular rings. We define δ(R) by δ(R)/Soc(RR) = Jac(R/Soc(RR)) and show, among others, the following results: (1) δ(R) is the largest δ-small right ideal of R. (2) R is δ-semiregular if and only if R/δ(R) is a von Neumann regular ring and idempotents of Rδ(R) lift to idempotents of R. (3) R is δ-semiperfect if and only if R/δ(R) is a semisimple ring and idempotents of R/δ(R) lift to idempotents of R. (4) R is δ-perfect if and only if R/Soc(RR) is a right perfect ring and idempotents of R/δ(R) lift to idempotents of R.

On Semiregular Rings whose Finitely Generated Modules Embed in Free Modules

AbstractWe consider rings as in the title and find the precise obstacle for them not to be Quasi-Frobenius, thus shedding new light on an old open question in Ring Theory. We also find several partial affirmative answers for that question.