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.

In this paper a ring means a commutative ring with identity element and S is a multiplicatively closed subset of the studied ring whose elements are regular. Inspiring from the work of J. Arnold about strongly finite type rings and D. E. Rush , s about noetherian spectrum rings, we introduce two types of rings that are S-strongly finite type rings and S-noetherian spectrum rings. We give some characterizations of these rings and we illustrate them by many examples.

Some properties like factoriality, seminormality and being a Krull domain, . . . are studied on power series rings AtXu 3 , AtXu 2 and AtXu 1 over a commutative ring A. If X is an uncountable set, there is an other sub-ring of AtXu 3 that stands strictly between AtXu 2

An elementary annihilator of a ring is an annihilator that has the form (0 : ) ; ∈ \ (0). We define the elementary annihilator dimension of the ring , denoted by EAdim( ), to be the upper bound of the set of all integers such that there is a chain (0 : 0 ) ⊂ ⋅ ⋅ ⋅ ⊂ (0 : ) of annihilators of . We use this dimension to characterize some zero-divisors graphs.

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