A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

Local rings whose modules are almost serial

The present paper is a sequel to our previous work on almost uniserial rings and modules, which appeared in the Journal of Algebra in 2016; it studies rings over which every (left and right) module is almost serial. A module is almost uniserial if any two of its submodules are either comparable in inclusion or isomorphic. And a module is almost serial if it is a direct sum of almost uniserial modules. The results of the paper are inspired by a characterization of Artinian serial rings as rings having all left (or right) modules serial. We prove that if R is a local ring and all left R-modules are almost serial then R is an Artinian ring which is uniserial either on the left or on the right. We also produce a connection between local rings having all left and right modules almost serial, local balanced rings studied by Dlab and Ringel and local Köthe rings. Finally we prove Morita invariance of the almost serial property and list some consequences.     

ON SOME SPECIAL CLASSES OF PRIME RINGS

Abstract

Given a non-zero cardinal α, a ring R is said to be SP(α) if a is the first cardinal for which every non-zero element of R has an insulator of cardinality less than α + 1.

It is shown that the class of SP(α) rings is a special class (in the sense of Andrunakievi?) for each α. A theorem of Groenewald and Heyman (also Desale and Varadarajan) to the effect that the class of all strongly prime rings is a special class is obtained as a corollary. Every SP(α) ring has an SP(α) rational extension ring with identity.     

CLASSICAL QUOTIENT RINGS OF GROUP RINGS

Abstract

Throughout G will denote a free Abelian group and Z(R) the right singular ideal of a ring R. A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R/Z(R) is semiprime, Z(R) is rationally closed, and Z(R) contains no closed uniform right ideals. We prove that R is a Cl-ring if and only if the group ring RG is a C1-ring. If RG has the additional property that bRG is dense whenever b is a right nonzero-divisor, then the complete ring of quotients of RG is a classical ring of quotients.     

STRONGLY PRIME GROUP RINGS

Abstract

A ring R is (right) strongly prime (SP) if every nonzero two sided ideal contains a finite set whose right annihilator is zero, SP rings have been studied by Handelman and Lawrence who raised the problem of characterizing SP group algebras. They showed that if R is SP and G is torsion free Abelian, then the group ring RG is SP. The aim of this note is to determine some more group rings which are SP.

For a ring R we also define the strongly prime radical s(R). We then show that s(R)G = s(W) for certain classes of groups.     

A NOTE ON LEFT STROUGLY NIL AND LEFT STROUGLY NILPOTENT

Abstract

In [2] van der Walt called a left ideal L of a ring A, left strongly nil, if given 1 ε L and k ε K, K a left ideal. there is an n such that (1+k)ⁿ ε K. L is called left strongly nilpotent if for any left ideal K there exists an m such that (L+K)^m ? K. In this paper we will prove that if A is a left artinian ring (not necessarily with unity) then every left strongly nil left ideal is left strongly nilpotent. This result is a generalization of the main theorem of [2].     

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.     

Quasi-duo rings and stable range descent

In a recent paper, the first author introduced a general theory of corner rings in noncommutative rings that generalized the classical theory of Peirce decompositions. This theory is applied here to the study of the stable range of rings upon descent to corner rings. A ring is called quasi-duo if every maximal 1-sided ideal is 2-sided. Various new characterizations are obtained for such rings. Using some of these characterizations, we prove that, if a quasi-duo ring R has stable range ?n, the same is true for any semisplit corner ring of R. This contrasts with earlier results of Vaserstein and Warfield, which showed that the stable range can increase unboundedly upon descent to (even) Peirce corner rings.     

HILBERT BASIS QUIVERS

Abstract

A quiver G (= directed multigraph, loops and parallel edges are allowed) is called a Hilbert basis quiver (HBQ) if a certain path algebra R[G] over a ring R is right noetherian provided R does. Such path algebras can be considered as generalized polynomial rings over R. There is the following characterization:

A quiver with a finite number of vertices is HBQ iff its set of edges is finite and its nontrivial path components are elementary cycles, up to parallel edges, which in addition are sink sets (i.e. there is no path leaving the component).

To prove this categorical methods are used.     

(n,d)-Injective covers,n-coherent rings,and (n,d)-rings

It is known that a ring R is left Noetherian if and only if every left R-module has an injective (pre)cover. We show that (1) if R is a right n-coherent ring, then every right R-module has an (n, d)-injective (pre)cover; (2) if R is a ring such that every (n, 0)-injective right R-module is n-pure extending, and if every right R-module has an (n, 0)-injective cover, then R is right n-coherent. As applications of these results, we give some characterizations of (n, d)-rings, von Neumann regular rings and semisimple rings.     

ON ANTISIMPLE GAMMA RINGS

ABSTRACT

In this note, we define the antisimple radical, A(M), of a Γ-ring M. A(M) is shown to be a special radical, and two characterizations of antisimple rings due to Szész are extended to Γ-rings. If R is the right operator ring of M, then A(R)* = A(M), where A(R) is the antisimple radical of R. If m,n are positive integers, then A(M_mn) = (A(M))_mn, where M_mn denotes the group m x n matrices over M, considered as a Γ_nm -ring with the operations of matrix addition and multiplication.     

每一个元素都是左零因子之环

本文引进左(右)零因子环的概念,它们是一类无单位元的环.我们称一个环为左(右)零因子环,如果对于任何 $a \in R$,都有$r_R (a) \neq 0~(l_R(a)\neq 0)$,而称一个环为强左(右)零因子环,如果$r_R(R)\neq 0~(l_R(R)\neq 0)$.Camillo和Nielson称一个环$R$为右有限零化环(简称RFA-环),如果$R$的每一个有限子集都有非零的右零化子.本文给出左零因子环的一些基本例子,探讨强左零因子环和RFA-环的扩张,并给出它们的等价刻画.     

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.     

On nonsingular right fpf rings

A right FPF ring is one over which every finitely generated faithful right module is a generator. The main purpose of the article is to givp the following cnaracterization of certain right FPF rings. TheoremLet R be semiprime and right semihereditary. Then R is right FPF iff (1) the right maximal ring of quotients Q_r (R) = Q coincides with the left and right classical rings of quotients and is self-injective regular of bounded index, (ii) R and Q have the same central idem-potents, (iii) if I is an ideal of R generated by a ma­ximal ideal of the boolean algebra of central idempotent s5 R/I is such that each non-zero finitely generated right ideal is a generator (hence prime), and (iv) R is such that every essential right ideal contains an ideal which is essential as a right ideal

In case that R is semiprime and module finite over its centre C, then the above can be used to show that R is FPF (both sides) if and only if it is a semi-hereditary maximal C-order in a self-injective regular ring (of finite index)

In order to prove the above it is shown that for any semiprime right FPF ring R, Q _lcl (R) exists and coincides with Q_r(R) (Faith and Page have shown that the latter is self-injective regular of bounded index). It R is semiprime right FPF and satisfies a polynamical identity then the factor rings as in (iii) above are right FPF and R is the ring of sections of a sheaf of prime right FPF rings

The Proofs use many results of C. Faith and S Page as well as some of the techniques of Pierce sheaves     

On the exactness of flat resolvents

Let R be a left coherent ring, FP — idRR the FP — injective dimension of RR and wD(R) the weak global dimension of R. It is shown that 1) FP -idRR < n ( n > 0) if and only if every flat resolvent 0 → M → F° → F¹... of a finitely presented right R—module M is exact at F'(i > n?1) if and only if every nth F -cosyzygy of a finitely presented right R — module has a flat preenvelope which is a monomorphism; 2) wD(R) < n (n > 1) if and only if every (n?l)th F-cosyzygy of a finitely presented right R—module has a flat preenvelope which is an epimorphism; 3) wD(R) 0) if and only if every nth F — cosyzygy of a finitely presented right R — module is flat. In particular, left FC rings and left semihereditary rings are characterized     

On a theorem of camillo

Let R denote a right principally injective ring. In this note we show that if R is right duo then R is right finite dimensional if and only if R has a finite number of maximal left ideals. This extends and answers an open question of Camillo. If, instead, every simple right module can be embedded in R, we show that R is left finite dimensional if it has a finite number of maximal right ideals.     

CHARACTERIZING THE NIL RADICAL OF A GAMMA RING

ABSTRACT

The nil radical, N(M) of a Γ-ring M was defined by Coppage and Luh [3], and shown by Groenewald [4] to be a special radical. We define s-prime ideals of M and show that N(M) is equal to the intersection of the s-prime ideals of M. If R is a ring, the nil radical of R considered as a Γ-ring with Γ = R is equal to the upper nil radical of R. We also give a sufficient condition for the equality N(R)* = N(M), where R is the right operator ring of M, and N(R) is its upper nil radical.     

Central semicommutative rings

A ring R is central semicommutative if ab = 0 implies that aRb ? Z(R) for any a, b ∈ R. Since every semicommutative ring is central semicommutative, we study sufficient condition for central semicommutative rings to be semicommutative. We prove that some results of semicommutative rings can be extended to central semicommutative rings for this general settings, in particular, it is shown that every central semicommutative ring is nil-semicommutative. We show that the class of central semicommutative rings lies strictly between classes of semicommutative rings and abelian rings. For an Armendariz ring R, we prove that R is central semicommutative if and only if the polynomial ring R[x] is central semicommutative. Moreover, for a central semicommutative ring R, it is proven that (1) R is strongly regular if and only if R is a left GP-V′-ring whose maximal essential left ideals are GW-ideals if and only if R is a left GP-V′-ring whose maximal essential right ideals are GW-ideals. (2) If R is a left SF and central semicommutative ring, then R is a strongly regular 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.     

WEAKLY CONTINUOUS AND C2-RINGS

A ring R is called right weakly continuous if the right annihilator of each element is essential in a summand of R, and R satisfies the right C2-condition (every right ideal that is isomorphic to a direct summand of R is itself a direct summand). We show that a ring R is right weakly continuous if and only if it is semiregular and J(R) = Z(R _R ). Unlike right continuous rings, these right weakly continuous rings form a Morita invariant class. The rings satisfying the right C2-condition are studied and used to investigate two conjectures about strongly right Johns rings and right FGF-rings and their relation to quasi-Frobenius rings.