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...

Different approaches to construct linear codes using skew polynomials can be unified by using the nonassociative algebras built from skew-polynomial rings by Petit.

Our purpose in this paper is to characterize skew PBW extensions over several weak symmetric rings. As a consequence of our treatment, we extend results the literature concerning property symmetry for commutative rings and polynomial

Let $R= \oplus_{ \alpha \in G} R_{\alpha}$ be a commutative ring with unity graded by an arbitrary grading monoid $G$. For each positive integer, the notions of graded-n-coherent module and are introduced. In this paper many results generalized from $n$-coherent rings to graded-$n$-coherent rings. last section, we provide necessary sufficient conditions for trivial extension graded-valuation ri...

A rst explicit connection between nitely presented commutative monoids and ideals in polynomial rings was used 1958 by Emelichev yielding a solution to the word problem in commutative monoids by deciding the ideal membership problem. The aim of this paper is to show in a similar fashion how congruences on monoids and groups can be characterized by ideals in respective monoid and group rings. Th...

For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring Tn(R,α). By using an ideal theory of a skew triangular matrix ring Tn(R,α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x;α]/〈xn〉, for each positive integer n, where R[x;α] is the skew polynomial ring, and 〈xn〉 is the ideal generated by xn.

In this paper we prove the universal property of skew PBW extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commutative rings. Skew PBW extensions include as particular examples Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Art...