Mc-Extensions: Examples, Zero-Divisors Graph, and Colorability

Abstract

Based on the well-known theorem of McCoy for polynomials and its generalization, by Roitman [14, Theorem 3.1], we introduce the concept of Mc-extension. An extension A ⊆ B of commutative rings is called an Mc-extension if for all subsets S of B such that there is a b ∈ B\ 0 satisfying bS = 0 then it exists an a ∈ A\ 0 satisfying aS = 0 . We study the transfer of some properties from one to the other member of an Mc-extension, using some examples like A ⊆ A 4 and /n ⊆ i /n i n ∈ .We compare their zero-divisor graphs. We study the colorability of B when A is a reduced coloring.

Rings of Formal Power Series in an Infinite Set of Indeterminates

“…In [4], Eljeri and Benhissi first introduced the ring D X ℵ 0 , where they denoted D X ℵ 0 by D X 4 , and they then in [3] studied some ringtheoretic properties of D X 4 . Among other things, they proved that (1) if for each finite n, D X 1 X n is a UFD then D X 4 is a UFD and (2) if D is a Krull domain then D X 4 is a Krull domain.…”

Some remarks on the ring

“…In [8], the authors introduced the sub-ring AtXu 4 = { +∞ i=0 F i ; F i ∈ AtXu 3 is zero or a form of degree i with countable support} of AtXu 3 . The identity element of AtXu k , k = 1, 2, 3 or 4, is the constant series which constant term is the identity element of A. AtXu i is a domain if and only if so is A.…”