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Abstract. In this paper we introduce the notion of weak zip rings and investigate their properties. We mainly prove that a ring R is right (left) weak zip if and only if for any n, the n-by-n upper triangular matrix ring T n (R) is right (left) weak zip. Let α be an endomorphism and δ an α-derivation of a ring R. Then R is a right (left) weak zip ring if and only if the skew polynomial ring R[x; α, δ] is a right (left) weak zip ring when R is (α, δ)-compatible and reversible.2000 MR Subject Classification. Primary 16S36, Secondary 16S99. [x; α, δ] as the Ore extension whose elements are the polynomials over R; the addition is defined as usual and the multiplication subject to the relation xa = α(a)x + δ(a) for any a ∈ R. Following Rage and Chhawchharia [14], a ring R is said to be Armendariz in that whenever polynomials f (x) = Introduction. Throughout this paper R denotes an associative ring with unity, α : R −→ R is an endomorphism and δ an α-derivation of R, that is, δ is an additive map such that δ(ab)

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Abstract. Let R be a ring and N il * (R) be the prime radical of R. In this paper, we say that a ring R is left N il * -coherent if N il * (R) is coherent as a left R-module. The concept is introduced as the generalization of left J-coherent rings and semiprime rings. Some properties of N il * -coherent rings are also studied in terms of N -injective modules and N -flat modules.