Commutative self-injective rings

Archiv der Mathematik -     

On regular rings and self-injective rings

This note contains certain conditions for unit-regularity and for selfinjective regular rings to have bounded index (such rings have many interesting properties [9]). Sufficient conditions are also given for rings to be Noetherian, Artinian and quasi-Frobeniusian.     

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring $R(\mathrm{\Lambda },V)$ and we describe $R(\mathrm{\Lambda },V)$ explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.     

On a generalisation of self-injective von Neumann regular rings

Apart from von Neumann regular rings, rings with infinite identities have not been studied in any detail. We take a first step in that direction by obtaining structure theorems for a class of self-injective rings with infinite identities. These extend the main structure theorems for self-injective von Neumann regular rings.

     

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.     

Regular rings and distributive lattices

We show that a finite lattice is isomorphic to the lattice of ideals in a regular ring if and only if it is distributive. We also study the countable case.     

Regular odd rings and non-planar graphs

In a previous paper we have announced that a graph is non-planar if and only if it contains a maximal, strict, compact, odd ring. Little has conjectured that the compactness condition may be removed. Chernyak has now published a proof of this conjecture. However, it is difficult to test a ring for maximality. In this paper we show that for odd rings of size five or greater, the condition of maximality may be replaced by a new one called regularity. Regularity is an easier condition to diagnose than is maximality.     

Weakly self-injective semilattices

In this note we characterize weakly self-injective semilattices as Brouwerian semilattices which are compact in the residuated interval topology. We also characterize weakly self-injective semigroups which are semilattices of groups with trivial multiplication. Research of first author supported in part by a research grant from the Faculty Research Committee of Bowling Green State University. Research of second author supported in part by a postdoctoral fellowship in the Biomathematics Program at North Carolina State University under PHS Grant #GM-678 from NIGMS.     

Recollements of self-injective algebras,and classification of self-injective diagram algebras

Mathematische Zeitschrift - Diagram algebras, in particular Brauer algebras, Birman–Murakami-Wenzl algebras and partition algebras, are used in representation theory and invariant theory of...