Abstract
Based on the well-known theorem of McCoy for polynomials and its generalization, by Roitman [
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, 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 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.