Isomorphism of generalized matrix rings

We give a constructive sufficient condition for a matrix over a commutative ring to be von Neumann regular, and we show that it is also necessary over local rings. Specifically, we prove that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible -submatrix if and only if the determinantal rank and the McCoy rank of A coincide. We deduce consequences to (products of local) commutative rings, and we determine the number of von Neumann regular matrices over some finite rings of residue classes and group algebras.

Let R be a commutative ring with identity. A short exact sequence of R-modules is said to be neat (respectively, -pure) if the sequence (respectively, ) is exact for every simple R-module S. The characterization of the integral domains over which neatness and -purity coincide has been given by László Fuchs: they are the integral domains where every maximal ideal is projective (and so also finitely generated). We show that for a commutative ring R, every maximal ideal of R is finitely generated and projective if and only if R has projective socle and neatness and -purity coincide. For a commutative ring R, we prove that neatness and -purity coincide if and only if every maximal ideal of R is finitely generated and the unique maximal ideal PP of the local ring RP is a principal ideal for every maximal ideal P of R. This result is proved firstly over commutative local rings and then using localization over any commutative ring. The Auslander-Bridger transpose of simple modules is used in proving these equivalences because it is an essential tool for the passage between proper classes of short exact sequences of modules that are projectively generated and these that are flatly generated by a set of finitely presented modules. For a commutative ring R, we prove that neatness and -purity coincide if and only if every simple R-module S is finitely presented and an Auslander-Bridger transpose of S is projectively equivalent to S.

Strong cleanness of the [formula omitted] matrix ring over a general local ring

A computer-aided study of the graded Lie algebra of a local commutative noetherian ring

Partial actions of groups on generalized matrix rings

Two questions on semi-clean group rings

A commutative local ring A of global dimension two conforms to the following description: (a) If the maximal ideal M is either principal or not finitely generated then A is a valuation domain, (b) Otherwise M is generated by a regular sequence of two elements but the ring is not necessarily noetherian.It will be noetherian if and only if it is completely integrally closed.

Products of commutators of transvections over local rings

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let ? be a Galois ring of characteristic 4,X a cyclotomic scheme over ? with classD and ? the Terwilliger algebra ofX, then we show that most of the irreducible ?-modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible ?-modules using Jacobi sums.

Berkani and Sarihr [Studia Math. (2001) 148: 251-257] showed that the set of all Drazin invertible elements in an algebra over a filed is a regularity in the sense of Kordula and M?ller [Studia Math. (1996) 119: 109-128]. In this paper, the above result is extended to the case of a ring. Counterexamples are provided to show that the set of all generalized Drazin invertible elements in a ring need not be a regularity in general. We determine when the set of all generalized Drazin invertible matrices in the 2 ? 2 full matrix ring over a commutative local ring is a regularity. We also give a sufficient condition for the set of all generalized Drazin invertible elements in a ring to be a regularity.

We establish that the derived Witt group is isomorphic to the usual Witt group when 2 is invertible. This key result opens the Ali Baba's cave of triangular Witt groups, linking the abstract results of Part I to classical questions for the usual Witt group. For commercial purposes, we survey the future applications of triangular Witt groups in the introduction. We also establish a connection between odd-indexed Witt groups and formations. Finally, we prove that over a commutative local ring in which 2 is a unit, the shifted derived Witt groups are all zero but the usual one.

Given a commutative local ring ( R , m ) (R,\mathfrak m) and an ideal I I of R R , a family of quotients of the Rees algebra R [ I t ] R[It] has been recently studied as a unified approach to the Nagata’s idealization and the amalgamated duplication and as a way to construct interesting examples, especially integral domains. When R R is noetherian of prime characteristic, we compute the Hilbert-Kunz function of the members of this family and, provided that either I I is m \mathfrak {m} -primary or R R is regular and F-finite, we also find their Hilbert-Kunz multiplicity. Some consequences and examples are explored.

Pseudocompact refinements of compact ring topologies

A Commutative Local Ring with Finite Global Dimension and Zero Divisors

We show that there is no nontrivial group homomorphism [Formula: see text] over commutative local rings and division rings for [Formula: see text], respectively. It gives a negative answer to Ye’s problem (see [S. K. Ye, Low-dimensional representations of matrix group actions on CAT(0) spaces and manifolds, J. Algebra 409 (2014) 219–243]) for the above rings.