Greg Marks

ON 2-PRIMAL ORE EXTENSIONS

When R is a local ring with a nilpotent maximal ideal, the Ore extension R[x; σ, δ] will or will not be 2-primal depending on the δ-stability of the maximal ideal of R. In the case where R[x; σ, δ] is 2-primal, it will satisfy an even stronger condition; in the case where R[x; σ, δ] is not 2-primal, it will fail to satisfy an even weaker condition.

Reversible and symmetric rings

Abstract We determine the precise relationships among three ring-theoretic conditions: duo , reversible , and symmetric . The conditions are also studied for rings without unity, and the effects of adjunction of unity are considered.

A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS

Let R be a ring, S a strictly ordered monoid, and ω : S→ End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the (S, ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S, ω)Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S, ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal. 2000 Mathematics subject classification: primary 16S99, 16W60; secondary 06F05, 16P60, 16S36, 16U80, 20M25.

On Unit-Central Rings

We establish commutativity theorems for certain classes of rings in which every invertible element is central, or, more generally, in which all invertible elements commute with one another. We prove that if R is a semiexchange ring (i.e., its factor ring modulo its Jacobson radical is an exchange ring) with all invertible elements central, then R is commutative. We also prove that if R is a semiexchange ring in which all invertible elements commute with one another, and R has no factor ring with two elements, then R is commutative. We offer some examples of noncommutative rings in which all invertible elements commute with one another, or are central. We close with a list of problems for further research.

Asymmetry in the Converse of Schur's Lemma

We show that the converse of Schurs Lemma can hold in the category of right modules, but not the category of left modules, over an appropriate ring. We exhibit classes of rings over which this left-right asymmetry does not occur, and provide new constructions of rings over whose module categories the converse of Schurs Lemma holds. We propose various open problems and avenues for further research concomitant to our work.

Extensions of Simple Modules and the Converse of Schur’s Lemma

The converse of Schur’s lemma (or CSL) condition on a module category has been the subject of considerable study in recent years. In this note we extend that work by developing basic properties of module categories in which the CSL condition governs modules of finite length.