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[[𝕏]]₄ and ℤ/nℤ ⊆ ℤ[i]/nℤ[i], n ∈ ℕ. We compare their zero-divisor graphs. We study the colorability of B when A is a reduced coloring.

Key Words:

• Colorings
• Formal power series rings
• Generalized formal power series rings
• Mc-extension
• Zero-divisor graph

2000 Mathematics Subject Classification:

• 13A15
• 13A99
• 13B99

Notes

Communicated by I. Swanson.