On Z-Symmetric Rings

We introduce the concept of Z-symmetric rings. In fact, the classes of all semicommutative rings, nil rings, reduced rings, Artinian rings and eversible rings are Z-symmetric rings. In order to sustain our assertion, we provide a number of examples of Z-symmetric and non Z-symmetric rings. We observe that the class of Z-symmetric rings lies strictly between the classes of eversible rings and the Dedekind finite rings. In particular, we consider the extensions of Z-symmetric rings. Finally, some new results between the Z-symmetric rings and Armendariz rings will be explored and investigated.

Quasi-central Armendariz rings

A ring [Formula: see text] is said to be quasi-central Armendariz if [Formula: see text] and [Formula: see text] satisfy [Formula: see text] then [Formula: see text] for all [Formula: see text] and [Formula: see text]. It is proved that if [Formula: see text] is a quasi-central Armendariz ring then [Formula: see text] implies that all [Formula: see text] are in its Wedderburn radical [Formula: see text], generalizing and improving the existing result in the literature.

Skew Hurwitz serieswise quasi-armendariz rings

For a ring endomorphism [Formula: see text], a generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of skew Hurwitz series type (or simply, [Formula: see text]-[Formula: see text]), is introduced and studied. It is shown that the [Formula: see text]-rings are closed upper triangular matrix rings, full matrix rings and Morita invariance. Some characterizations for the skew Hurwitz series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and semiprime are concluded.

On Almost Armendariz Ring

In this paper, we introduce the notion of an almost Armendariz ring, which is a generalization of an Armendariz ring, and discuss some of its properties. It has been found that every almost Armendariz ring is weak Armendariz but the converse is not true. We prove that a ring R is almost Armendariz if and only if R[x] is almost Armendariz. It is also shown that if R/I is an almost Armendariz ring and I is a semicommutative ideal, then R is an almost Armendariz ring. Moreover, the class of minimal non-commutative almost Armendariz rings is completely determined, up to isomorphism (minimal means having smallest cardinality).

Extension of Antoine’s Ring Construction

This article concerns a kind of extension of Antoine’s ring construction which is unit-IFP and Armendariz. Such extensions are also shown to be unit-IFP and Armendariz, by which we may extend the classes of unit-IFP and Armendariz rings.