Some Characterizations of VNL Rings

A ring R is said to be von Newmann local (VNL) if for any a ∈ R, either a or 1 ?a is (von Neumann) regular. The class of VNL rings lies properly between exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without an infinite set of orthogonal idempotents; and also the VNL rings having a primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a ₁, a ₂) ∈ R ², one of the a _i 's is regular in R. Formal triangular matrix rings that are VNL are also characterized. As a corollary, it is shown that an upper triangular matrix ring T _n (R) is VNL if and only if n = 2 or 3 and R is a division ring.     

EXTENSIONS OF CLEAN RINGS

It is shown that if e is an idempotent in a ring R such that both eRe and (1 ? e)R(1 ? e) are clean rings, then R is a clean ring. This implies that the matrix ring M _n (R) over a clean ring is clean, and it gives a quick proof that every semiperfect is clean. Other extensions of clean rings are studied, including group rings.     

On Weak Armendariz Rings

We introduce weak Armendariz rings which are a generalization of semicommutative rings and Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak Armendariz if and only if for any n, the n-by-n upper triangular matrix ring T _n (R) is weak Armendariz. If R is semicommutative, then it is proven that the polynomial ring R[x] over R and the ring R[x]/(x ⁿ ), where (x ⁿ ) is the ideal generated by x ⁿ and n is a positive integer, are weak Armendariz.     

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 Classification of Commutative Reduced Filial Rings

A ring R is said to be filial when for every I, J, if I is an ideal of J and J is an ideal of R then I is an ideal of R. The classification of commutative reduced filial rings is given.     

When is a Sum of Annihilator Ideals an Annihilator Ideal?

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.     

Presimplifiable and Cyclic Group Rings

Let R be any commutative ring with identity, and let C be a (finite or infinite) cyclic group. We show that the group ring R(C) is presimplifiable if and only if its augmentation ideal I(C) is presimplifiable. We conjecture that the group rings R(C _n ) are presimplifiable if and only if n = p ^m , p ∈ J(R), p is prime, and R is presimplifiable. We show the necessity of n = p ^m , and we prove the sufficiency when n = 2, 3, 4. These results were made possible by a new formula derived herein for the circulant determinantal coefficients.     

Condensed Rings with Zero-Divisors #

A commutative ring R with identity is condensed (respectively strongly condensed) if for each pair of ideals I, J of R, IJ = {ij | i ∈ I, j ∈ J} (resp., IJ = iJ for some i ∈ I or IJ = Ij for some j ∈ J). In a similar fashion we can define regularly condensed and regularly strongly condensed rings by restricting I and J to be regular ideals. We show that an arbitrary product of rings is condensed if and only if each factor is so, and that R[X] is condensed if and only if R is von Neumann regular. A number of results known in the domain case are extended to the ring case. Regularly strongly condensed and one-dimensional regularly condensed Noetherian rings are characterized.     

On the quotient between length and multiplicity

In this paper, the exchange ring R with the (general) ?₀-comparability is studied. A ring R is said to satisfy the general ?₀-comparability, if for any idempotent elements f, g ∈ R, there exist a positive integer n and a central idempotent element e ∈ R such that f Re ?_⊕ n[gRe] and gR(1 ? e) ?_⊕ n[f R(1 ? e)]. It is proved that the (general) ?₀-comparability for exchange rings is preserved under taking factor rings, matrix rings and corners. The ?₀-comparability condition for exchange rings R is characterized by the order structure of several partially ordered sets of ideals of R. For any exchange ring R with general ?₀-comparability and any proper ideal I of R not contained in J(R), it is proved that if I contains no nonzero central idempotents of R, then: 1) There exists an infinite set of nonzero idempotent elements {f _i ∣ i = 1,2, …} in I such that f ₁ R ?_⊕ f ₂ R ?_⊕ …, and n(f _n R) ?_⊕ R _R for all n ≥ 1; 2) For any m ≥ 1, there exist nonzero orthogonal idempotents e ₁, e ₂ …, e _m in I such that e ₁ R ⊕ e ₂ R ⊕ … ⊕ e _m R ?_⊕ I _R and e _i R ? e _j R for all i, j. For any exchange ring R with primitive factor rings artinian, if R satisfies the general ?₀-comparability, then in every ideal I of R not contained in J(R), there is a central idempotent element of R.     

Free Pairs of Bicyclic Units in the Integral Group Ring of the Dihedral Group

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.     

Minimal But Inefficient Presentations for Self Semidirect Products of the Free Abelian Monoid on Two Generators

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.     

Power-substitution,exchange rings and unit II-regularity

In this paper,we investigate power-substitution over exchange rings.We show that an exchange ring R satisfies power-substitution if and only if for any regular x ∈ R, there exists a positive integer n such that xI _n is unit πregular in M _n(R).     

On Generalizations of Commutativity

This note is concerned with generalizations of commutativity. We introduce identity-symmetric and right near-commutative, and study basic structures of rings with such ring properties. It is shown that if R is an identity-symmetric ring, then the set of all nilpotent elements forms a commutative subring of R. Moreover, identity-symmetric regular rings are proved to be commutative. The near-commutativity is shown to be not left-right symmetric, and we study some conditions under which the near-commutativity is left-right symmetric. We also examine the near-commutativity of skew-trivial extensions, which has a role in this note.     

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. Received February 20, 2001, Accepted July 20, 2001     

Armendariz Properties Relative to a Ring Endomorphism

For an endomorphism α of a ring R, we introduce the notion of an α-Armendariz ring to investigate the relative Armendariz properties. This concept extends the class of Armendariz rings and gives us an opportunity to study Armendariz rings in a general setting. It is obvious that every Armendariz ring is an α-Armendariz ring, but we shall give an example to show that there exists a right α-Armendariz ring which is not Armendariz. A number of properties of this version are established. It is shown that if I is a reduced ideal of a ring R such that R/I is a right α-Armendariz ring, then R is right α-Armendariz. For an endomorphism α of a ring R, we show that R is right α-Armendariz if and only if R[x] is right α-Armendariz. Moreover, a weak form of α-Armendariz rings is considered in the last section. We show that in general weak α-Armendariz rings need not be α-Armendariz.     

A Note on rings of finite rank

The rank rk(R) of a ring R is the supremum of minimal cardinalities of generating sets of I as I ranges over ideals of R. Matsuda and Matson showed that every n∈?⁺ (the positive integers) occurs as the rank of some ring R. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension 0 or 1, we give four different constructions of rings of rank n (for all n∈?⁺). Two constructions use one-dimensional domains. Our third construction uses Artinian rings (dimension zero), and our last construction uses polynomial rings over local Artinian rings (dimension one, irreducible, not a domain).     

Generalized Derivations with Nilpotent Values on Semiprime Rings

Let R be a semiprime ring, RF be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of RE. Furthermore, if there exists a fixed positive integer n such that μ(x)^n = 0 for all x∈I, then μ=0.     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

Indecomposable Modules and Gelfand Rings

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R _P is an Artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings. It is shown that each commutative ring R with a Hausdorff and totally disconnected maximal spectrum is local-global. Moreover, if R is arithmetic, then R is an elementary divisor ring.     

On commutative rings which are strongly prüfer

Abstract

As defined by Nicholson [Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278] an element of a ring R is clean if it is the sum of a unit and an idempotent, and a subset A of R is clean if every element of A is clean. It is shown that a semiprimitive Gelfand ring R is clean if and only if Max(R) is zero-dimensional; if and only if for each M ∈ Max(R), the intersection all prime ideals contained in M is generated by a set of idempotents. We also give several equivalent conditions for clean functional rings. In fact, a functional ring R is clean if and only if the set of clean elements is closed under sum; if and only if every zero-divisor is clean; if and only if; R has a clean prime ideal.