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.     

Regular semiartinian rings

We study the structure of rings over which every right module is an essential extension of a semisimple module by an injective one. A ring R is called a right max-ring if every nonzero right R-module has a maximal submodule. We describe normal regular semiartinian rings whose endomorphism ring of the minimal injective cogenerator is a max-ring.     

On normal classes of rings

In [6] Lanski,Resco and Small proved that, if I is a non-zero right ideal of a prime ring R then R is right primitive if and only if I is right primitive modulo its prime radical. Considering the opposite ring one gets the left version of this result. It is natural to ask whether the mixed version C left ideal, right primitivity) of the theorem holds as well. Studying this question we concluded that all the results of [6] can be extended to normal classes of rings[7] (of which the classes of left and right primitive rings are examples). It in particular answers positively the question. we also get several new characterizations of normal classes and find a direct proof of the quoted result of [6].     

Nil-Clean Rings with Involution

A *-ring R is called a nil *-clean ring if every element of R is a sum of a projection and a nilpotent.Nil *-clean rings are the *-version of nil-clean rings introduced by Diesl.This paper is about the nil *-clean property of rings with emphasis on matrix rings.We show that a *-ring R is nil *-clean if and only if J(R) is nil and R/J(R) is nil*-clean.For a 2-primal *-ring R,with the induced involution given by (aij)* =(a*ij)T,the nil *-clean property of Mn(R) is completely reduced to that of Mn(Z2).Consequently,Mn(R) is not a nil *-clean ring for n =3,4,and M2(R) is a nil *-clean ring if and only if J(R) is nil,R/J(R) is a Boolean ring and a*-a ∈ J(R) for all a ∈ R.     

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension.

In this article, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.     

与线性变换的完全环同构的环理论(Ⅱ)

<正> 为了进一步对本原环结构的研究,本文引进规范环的概念,我们说环R是规范的,若R是一个线性变换完全环并且及的基座对于任一对应基{E_i}皆有=∑RE_i=∑E_iR.容易知道,满足单侧理想极小条件的单纯环必是规范的.     

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b? R with aR + bR = R, there exists y ? R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ? R), Theorem 6: A left or right N ₀o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N ₀-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N _o-continuous von Neumann regular ring is unit-regular     

Quasi-Frobenius Rings and Nakayama Permutations of Semiperfect Rings

We say that is a ring with duality for simple modules, or simply a DSM-ring, if, for every simple right (left) -module U, the dual module U* is a simple left (right) -module. We prove that a semiperfect ring is a DSM-ring if and only if it admits a Nakayama permutation. We introduce the notion of a monomial ideal of a semiperfect ring and study the structure of hereditary semiperfect rings with monomial ideals. We consider perfect rings with monomial socles.     

On finitely embedded rings

A result of Ginn and Moss asserts that a left and right noetherian ring with essential right socle is left and right artinian. There are examples of right finitely embedded rings with ACC on left and right annihilators which are not artinian. Motivated by this, it was shown by Faith that a commutative, finitely embedded ring with ACC on annihilators (and square-free socle) is artinian (quasi-Frobenius). A ring R is called right minsymmetric if, whenever k R is a simple right ideal of R, then R k is also simple. In this paper we show that a right noetherian right minsymmetric ring with essential right socle is right artinian. As a consequence we show that a ring is quasi-Frobenius if and only if it is a right and left mininjective, right finitely embedded ring with ACC on right annihilators. This extends the known work in the artinian case, and also extends Faith's result to the non-commutative case.     

Rings Whose Modules Have Grade Zero

In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.     

Uniserial Noetherian centrally essential rings

Abstract

It is proved that a ring R is a right uniserial, right Noetherian centrally essential ring if and only if R is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings.     

F q [M 2], F q [GL 2], and F q [SL 2] as Quantized Hyperalgebras

A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M _n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.     

FP-RINGS

A ring R is called right FP-injective if every R-homomor-phism from a finitely generated submodule of a free right R-module F into R extends to F. In this paper a ring R will be called a right FP-ring if R is semiperfect, right FP-injective and has an essential right socle. The class of FP-rings strictly contains the class of right (and left) pseudo-Frobenius rings, and we show that it is right-left symmetric and Morita-invariant. As an application we show that if R is a left perfect right FP-injective ring, then R is quasi-Frobenius if and only if the second right socle of R is finitely generated as a right ideal of R. This extends the known results in the right selfinjective case.     

RINGS WHOSE SIMPLE MODULES ARE ABSOLUTELY PURE

A ring is called right SAP if every right simple module over it is absolutely pure. In this paper we prove that every right SAP ring is semiprimitive and that the homomorphic image and the center of an right SAP ring are also right SAP. We also show that the sum of all absolutely pure minimal submodules of any module is a fully invariant submodule. As an application, we give a decomposition of some selfinjective rings.     

Generalized APP-Rings

We say a ring R is (centrally) generalized left annihilator of principal ideal is pure (APP) if the left annihilator ? _R (Ra) ⁿ is (centrally) right s-unital for every element a ∈ R and some positive integer n. The class of generalized left APP-rings includes generalized left (principally) quasi-Baer rings and left APP-rings (and hence left p.q.-Baer rings, right p.q.-Baer rings, and right PP-rings). The class of centrally generalized left APP-rings is closed under finite direct products, full matrix rings, and Morita invariance. The behavior of the (centrally) generalized left APP condition is investigated with respect to various constructions and extensions, and it is used to generalize many results on generalized PP-rings with IFP and semiprime left APP-rings. Moreover, we extend a theorem of Kist for commutative PP rings to centrally generalized left APP rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Furthermore, we give a complete characterization of a considerably large family of centrally generalized left APP rings which have a sheaf representation.     

PRIME IDEALS IN RINGS WITH INVOLUTION

Abstract

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.     

广义FP—内射模、广义平坦模与某些环

左（右）R－模A称为GFP－内射模，如果ExtR(M,A)=0对任－2－表现R－模M成立；左（右）R－模称为G－平坦的，如果Tor1^R(M,A)=0(Tor1^R(AM)=0)对于任一2－表现右（左）R－模M成立；环R称左（右）R－半遗传环，如果投射左（右）R－模的有限表现子模是投射的，环R称为左（右）G－正而环，如果自由左（右）R－模的有限表现子模为其直和项，研究了GFP－内射模和G－平坦模的一些性质，给出了它们的一些等价刻划，并利用它们刻划了凝聚环，G－半遗传环和G－正则环。     

Left SF-Rings and Regular Rings

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.     

Star-group identities and groups of units

Analogous to *-identities in rings with involution we define *-identities in groups. Suppose that G is a torsion group with involution * and that F is an infinite field with char F ≠ 2. Extend * linearly to FG. We prove that the unit group U{\mathcal{U}} of FG satisfies a *-identity if and only if the symmetric elements U⁺{\mathcal{U}^+} satisfy a group identity.     

*-Regular Leavitt path algebras of arbitrary graphs

If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L _K (E). We show that the involution on L _K (E) is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L _K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L _K (E) to be *-regular (i.e., regular with proper involution). This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone. As a corollary, we show that Handelman’s conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.