Undirected Zero-Divisor Graphs and Unique Product Monoid Rings

Let R be an associative ring with identity and Z*(R) be its set of non-zero zero-divisors. The undirected zero-divisor graph of R, denoted by Γ(R), is the graph whose vertices are the non-zero zero-divisors of R, and where two distinct vertices r and s are adjacent if and only if rs = 0 or sr = 0. The distance between vertices a and b is the length of the shortest path connecting them, and the diameter of the graph, diam(Γ(R)), is the superimum of these distances. In this paper, first we prove some results about Γ(R) of a semi-commutative ring R. Then, for a reversible ring R and a unique product monoid M, we prove 0≤ diam(Γ(R))≤ diam(Γ(R[M]))≤3. We describe all the possibilities for the pair diam(Γ(R)) and diam(Γ(R[M])), strictly in terms of the properties of a ring R, where R is a reversible ring and M is a unique product monoid. Moreover, an example showing the necessity of our assumptions is provided.

Answers to Some Questions Concerning Rings with Property (A)

A ring R has right property (A) whenever a finitely generated two-sided ideal of R consisting entirely of left zero-divisors has a non-zero right annihilator. As the main result of this paper we give answers to two questions related to property (A), raised by Hong et al. One of the questions has a positive answer and we obtain it as a simple conclusion of the fact that if R is a right duo ring and M is a u.p.-monoid (unique product monoid), then R is right M-McCoy and the monoid ring R[M] has right property (A). The second question has a negative answer and we demonstrate this by constructing a suitable example.

Nilradicals of the unique product monoid rings

Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring [Formula: see text] and a monoid [Formula: see text], we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring [Formula: see text]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.

Extensions of strongly reflexive rings

This is a further study of reflexive rings over polynomial rings and monoid rings. The concepts of strongly reflexive rings and strongly [Formula: see text]-reflexive rings are introduced and investigated. Some characterizations of various extensions of the two classes of rings are obtained. It is proved that a ring [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive. For a right Ore ring [Formula: see text] with classical right quotient ring [Formula: see text], we show that [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive. Moreover, we prove that if [Formula: see text] is a unique product monoid (u.p.-monoid) and [Formula: see text] is a reduced ring, then [Formula: see text] is strongly [Formula: see text]-reflexive. It is shown that finite direct sums of strongly [Formula: see text]-reflexive rings are strongly [Formula: see text]-reflexive.

REVERSIBLE SKEW GENERALIZED POWER SERIES RINGS

In this note we show that there exist a semiprime ring R, a strictly ordered artinian, narrow, unique product monoid (S,≤) and a monoid homomorphism ω:S⟶End(R) such that the skew generalized power series ring R[[S,ω]] is semicommutative but R[[S,ω]] is not reversible. This answers a question posed in Marks et al. [‘A unified approach to various generalizations of Armendariz rings’, Bull. Aust. Math. Soc.81 (2010), 361–397].