We investigate non-unique factorization of polynomials in Zpn [x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Zpn [x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of Zpn [x]...

Let be a Banach algebra and : a derivation. In this paper, it is proved, under certain conditions, that , where is the Jacobson radical of . Moreover, we prove that if is unital and : is a continuous derivation, then ⋂ ⋂ ⋂ , where denotes the set of all primitive ideals such that is commutative, denotes the set of all maximal (modular) ideals such that is commutative, and Φ is the set of all no...

Throughout, R will be a commutative ring with identity with total quotient ring T (R), group of units U(R), set of zero-divisors Z(R), and Jacobson radical J(R). For a; b 2 R, we de...ne three associate relations: 1. We say a and b are associate, denoted a » b, if ajb and bja, (a) = (b). 2. We say a and b are strongly associate, denoted a 1⁄4 b, if there exists a u 2 U (R) such that a = ub. 3. ...

It has been conjectured that the group algebra Q[G] of a discrete group G over the rational numbers Q is semi-simple. In this paper we consider irreducible representations of this algebra satisfying certain finiteness conditions and see just how much information these yield towards solving the above problem. We show in fact for a finitely generated group G, Q[G] has "enough" of these representa...

We study the structure of 0-primitive near-rings and are able to answer an open question in the theory of minimal ideals in near-rings to the negative, namely if the heart of a zero symmetric subdirectly irreducible near-ring is subdirectly irreducible again. Also, we will be able to classify when a simple near-ring with an identity and containing a minimal left ideal is a Jacobson radical near...

In this paper we describe an algorithm for finding the nilpotency class, and the upper central series of the maximal normal p-subgroup ∆(G) of the automorphism group, Aut(G) of a bounded (or finite) abelian p-group G. This is the first part of two papers devoted to compute the nilpotency class of ∆(G) using formulas, and algorithms that work in almost all groups. Here, we prove that for p ≥ 3 t...

Given an Artinian algebra A over a field k, there are several combinatorial objects associated to A. They are the diagram DA as defined by Drozd and Kirichenko, the natural quiver ΔA defined by Li (cf. Section 2), and a generalized version of k-species (A/r, r/r2) with r being the Jacobson radical of A. When A is splitting over the field k, the diagram DA and the well-known Ext-quiver ΓA are th...