The concept of a zero-divisor graph of a commutative ring was first introduced in Beck (1988), and later redefined in Anderson and Livingston (1999). Redmond (2002) further extended this concept to the noncommutative case, introducing several definitions of a zero-divisor graph of a noncommutative ring. Recently, the diameter and girth of polynomial and power series rings over a commutative rin...

We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local k-algebra R, not necessarily commutative, where k is a field. The class of topological modules we consider include all those of finite rank over k and some of infinite rank as well, namely those with a Schauder basis in the sense of §1. This generalizes the resul...

Abstract We prove asymptotics and study sign patterns for coefficients in expansions of elements the Habiro ring which satisfy a strange identity. As an application, we discuss positivity generalized Fishburn numbers arise from Kontsevich–Zagier series associated to colored Jones polynomial family torus knots. This extends Zagier’s result on numbers.

We discuss the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, formal power series, idealization of the R-module M and relations between the total graph of the ring R and its extensions are also dealt with.

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring Z[[x]] of formal power series with integer coefficients. For n,m ≥ 1 and p prime, we show that p+pβx+αx is reducible in Z[[x]] if and only if it is reducible in Zp[x], the ring of polynomials over the p-adic integers.

This paper first improves Chen and Hsieh’s definition of generalized fuzzy numbers, which makes it the generalization of definition of fuzzy numbers. Secondly, in terms of the generalized fuzzy numbers set, we introduce two different kinds of orders and arithmetic operations and metrics based on the λ-cutting sets or generalized λ-cutting sets, so that the generalized fuzzy numbers are integrat...

Let A be a semprime, right noetherian ring equipped with an automorphism α, and let B := A[[y;α]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.

Given an iterated skew polynomial ring C[y1; τ1, δ1] . . . [yn; τn, δn] over a complete local ring C with maximal ideal m, we prove, under suitable assumptions, that the completion at the ideal m+ 〈y1, y2, . . . , yn〉 is an iterated skew power series ring. When C is a field, this completion is a local, noetherian, Auslander regular domain with Krull, classical Krull and global dimension all equ...