n-ideal Near rings

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Exchange rings satisfying power ideal-cancellation

In this paper, we introduce a new kind of partial power cancellations, and get a number of necessary and sufficient conditions for exchange rings satisfying power ideal-cancellation. These also extend many known results.     

RadicalsofGronpG－gradedΓ－rings

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Type Ⅱ codes over finite rings

In this paper,we generalize the concept of Type Ⅱ codes to arbitrary finite rings. We focus on Type Ⅱ codes over finite chain rings and use the Chinese Remainder Theorem on these codes to study Type Ⅱ codes over principal ideal rings.     

Hilbert's theorem 90 in matrix rings †

The fact that every unitary matrix can be expressed in the form A^-1A^? (observed by C. R. Johnson) is interpreted as a "Hilbert Theorem 90" in rings with an involution. Several observations connected with this are included.     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.     

Direct limits of Cohen–Macaulay rings

Let A be a direct limit of a direct system of Cohen–Macaulay rings. In this paper, we describe the Cohen–Macaulay property of A. Our results indicate that A is not necessarily Cohen–Macaulay. We show A is Cohen–Macaulay under various assumptions. As an application, we study Cohen–Macaulayness of non-affine normal semigroup rings.     

Mod-retractable strongly π-regular rings and group algebras

Abstract

In this work, we show that a two-sided mod-retractable and strongly π-regular ring is two-sided semiartinian. On the other hand, a strongly π-regular von Neumann regular ring is left mod-retractable if and only if it is a left semiartinian V-ring. We apply these results to group algebras and thereby we give the complete structure of groups whose group algebras are mod-retractable.

Communicated by Toma Albu     

Algebraic games—playing with groups and rings

Two players alternate moves in the following impartial combinatorial game: Given a finitely generated abelian group A, a move consists of picking some \(0 \ne a \in A\). The game then continues with the quotient group \(A/\langle a \rangle \). We prove that under the normal play rule, the second player has a winning strategy if and only if A is a square, i.e. \(A \cong B \times B\) for some abelian group B. Under the misère play rule, only minor modifications concerning elementary abelian groups are necessary to describe the winning situations. We also compute the nimbers, i.e. Sprague–Grundy values of 2-generated abelian groups. An analogous game can be played with arbitrary algebraic structures. We study some examples of non-abelian groups and commutative rings such as R[X], where R is a principal ideal domain.     

Generalized inverses modulo ℋ in semigroups and rings

The definition of the inverse along an element was very recently introduced, and it contains known generalized inverses such as the group, Drazin and Moore–Penrose inverses. In this article, we first prove a simple existence criterion for this inverse in a semigroup, and then relate the existence of such an inverse in a ring to the ring units.     

Vanishing of cohomology over Cohen–Macaulay rings

A 2003 counterexample to a conjecture of Auslander brought attention to a family of rings—colloquially called AC rings—that satisfy a natural condition on vanishing of cohomology. Several results attest to the remarkable homological properties of AC rings, but their definition is barely operational, and it remains unknown if they form a class that is closed under typical constructions in ring theory. In this paper, we study transfer of the AC property along local homomorphisms of Cohen–Macaulay rings. In particular, we show that the AC property is preserved by standard procedures in local algebra. Our results also yield new examples of Cohen–Macaulay AC rings.     

Kötter interpolation in skew polynomial rings

Skew polynomials are a noncommutative generalization of ordinary polynomials that, in recent years, have found applications in coding theory and cryptography. Viewed as functions, skew polynomials have a well-defined evaluation map; however, little is known about skew-polynomial interpolation. In this work, we apply Kötter’s interpolation framework to free modules over skew polynomial rings. As a special case, we introduce a simple interpolation algorithm akin to Newton interpolation for ordinary polynomials.     

On noncommutative Gröbner bases over rings

Let R be a commutative ring. It is proved that for verification of whether a set of elements {f _α} of the free associative algebra over R is a Gröbner basis (with respect to some admissible monomial order) of the (bilateral) ideal that the elements f _α generate it is sufficient to check the reducibility to zero of S-polynomials with respect to {f _α} iff R is an arithmetical ring. Some related open questions and examples are also discussed.     

The structure of a class of Z-local rings

A local ring R is called Z-local if J(R) = Z(R) and J(R)2 = 0. In this paper the structure of a class of Z-local rings is determined.