Alternative regular rings without nilpotent elements

Nilpotent Elements in Skew Polynomial Rings

 Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We al...

The Model Companion of the Theory of Commutative Rings without Nilpotent Elements

We show that the theory of commutative rings without nilpotent elements has a model companion. The model companion is decidable and is the model completion of the theory of commutative regular rings. Recall that a theory K is model-complete if for any model M of K, KuD(M) is complete, where D(M) denotes the diagram of M. A natural generalization of this notion is that of a model completion. We ...

Commutative Regular Rings without Prime Model Extensions

It is known that the theory K of commutative regular rings with identity has a model completion K . We show that there exists a countable model of K which has no prime extension to a model of K'. If K and K ate theories in a first order language L, then K is said to be a model completion of K if K extends K, every model of K can be embedded in a model of K , and for any model A of K and models ...

Extensions of Nilpotent P.P. Rings

Commuting $pi$-regular rings

R is called commuting regular ring (resp. semigroup) if for each x,y $in$ R there exists a $in$ R such that xy = yxayx. In this paper, we introduce the concept of commuting $pi$-regular rings (resp. semigroups) and study various properties of them.