Abstract. We, in this paper, study the commutativity of skew polynomials at zero as a generalization of an α-rigid ring, introducing the concept of strongly skew reversibility. A ring R is be said to be strongly α-skew reversible if the skew polynomial ring R[x; α] is reversible. We examine some characterizations and extensions of strongly α-skew reversible rings in relation with several ring theoretic properties which have roles in ring theory.
In this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak Armendariz rings and quasi-Armendariz rings. It is shown that the weak quasi-Armendarizness is a Morita invariant property. For a semiprime ring R, it is shown that R[x]/〈xn〉 is weak quasi-Armendariz, where R[x] is the polynomial ring over R and 〈xn〉 is the ideal of R[x] generated by xn. Various properties of weak quasi-Armendariz rings are also observed.
Firstly, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules over an arbitrary ring. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined over algebras given by quivers.
Abstract. In this paper, we investigate the insertion-of-factors-property (simply, IFP) on skew polynomial rings, introducing the concept of strongly σ-IFP for a ring endomorphism σ. A ring R is said to have strongly σ-IFP if the skew polynomial ring R[x; σ] has IFP. We examine some characterizations and extensions of strongly σ-IFP rings in relation with several ring theoretic properties which have important roles in ring theory. We also extend many of related basic results to the wider classes, and so several known results follow as consequences of our results.
Abstract. For a ring endomorphism α of a ring R, Krempa called α a rigid endomorphism if aα(a) = 0 implies a = 0 for a ∈ R, and Hong et al. called R an α-rigid ring if there exists a rigid endomorphism α. Due to Rege and Chhawchharia, a ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., α-Armendariz rings and α-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between α-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an α-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.Throughout this paper, all rings are associative with identity. Given a ring R, the polynomial ring over R is denoted by R [x]. Recall that a ring R is called reduced if it has no nonzero nilpotent elements. Armendariz [1, Lemma 1] showed that for a reduced ring R, if any polynomial [2,3,5,6,8,10,11,12].The reducedness and Armendariz property of a ring were extended as follows. For a ring R with a ring endomorphism α : R → R, a skew polynomial ring (also called an Ore extension of endomorphism type) R[x; α] of R is the ring obtained by giving the polynomial ring over R with the new multiplication xr = α(r)x for all r ∈ R. Recall that an endomorphism α of a ring R is called rigid [9] if aα(a) = 0 implies a = 0 for a ∈ R, and a ring R is called α-rigid [4] if there exists a rigid endomorphism α of R. Note that any rigid endomorphism of a ring is a monomorphism, and α-rigid rings are reduced rings [4, Proposition 5]. On the other hand, the Armendariz property with respect to polynomials was extended to one of skew polynomials. A ring R is called α-Armendariz (resp.,
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