Almost zip Bezout domain

J. Zelmanowitz introduced the concept of a ring, which we call a zip ring. In this paper we characterize a commutative Bezout domain whose finite homomorphic images are zip rings modulo its nilradical.

On Annihilator Ideals of Pseudo-differential Operator Rings

Let R be a ring with a derivation δ and R((x-1; δ)) denote the pseudo-differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R((x-1; δ)). Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R((x-1; δ)) is Baer (resp., quasi-Baer, PP, right zip) if and only if R is a Baer (resp., quasi-Baer, PP, right zip) ring. For a δ-weakly rigid ring R, R((x-1; δ)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.

MAL'CEV–NEUMANN SERIES OVER ZIP AND WEAK ZIP RINGS

In this paper we show that: if G is a totally ordered group and R is a G-Armendariz ring (an NI ring with nil (R) is nilpotent), then the ring Λ = R((G; σ; τ)) of Mal'cev–Neumann series is a right zip (weak zip) ring if and only if R is.

RINGS WITH THE BEACHY–BLAIR CONDITION

A ring satisfies the left Beachy–Blair condition if each of its faithful left ideal is cofaithful. Every left zip ring satisfies the left Beachy–Blair condition, but both properties are not equivalent. In this paper we will study the similarities and the differences between zip rings and rings with the Beachy–Blair condition. We will also study the relationship between the Beachy–Blair condition of a ring and its skew polynomial and skew power series extensions. We give an example of a right zip ring that is not left zip, proving that the zip property is not symmetric.

Extensions of zip rings

A ring R is called right zip provided that if the right annihilator rR(X) of a subset X of R is zero, then there exists a finite subset Y of X, such that rR(Y) = 0. Faith [6] raised the following questions: When does R being a right zip ring imply R[x] being right zip?; When does R being a right zip imply R[G] being right zip when G is a finite group?; Characterize a ring R such that Matn(R) is right zip. In this note we continue the study of the extensions of non-commutative zip rings based on Faith’s questions. It is shown that if R is a right McCoy ring, then R is right zip if and only if R[x] is a right zip ring. Also, if M is a strictly totally ordered monoid and R a right duo ring or a reversible ring, then R is right zip if and only if R[M] is right zip. As a consequence we obtain a generalization of [7].

ORE EXTENSIONS OF WEAK ZIP RINGS

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 Tn(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.