Rings with S-acc on d-annihilators

A commutative ring [Formula: see text] is said to satisfy acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is stationary. In this paper we extend the notion of rings with acc on d-annihilators by introducing the concept of rings with [Formula: see text]-acc on d-annihilators, where [Formula: see text] is a multiplicative set. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] satisfies [Formula: see text]-acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is [Formula: see text]-stationary, that is, there exist a positive integer [Formula: see text] and an [Formula: see text] such that for each [Formula: see text] [Formula: see text] We give equivalent conditions for the power series (respectively, polynomial) ring over an Armendariz ring to satisfy [Formula: see text]-acc on d-annihilators. We also study serval properties of rings satisfying [Formula: see text]-acc on d-annihilators. The concept of the amalgamated duplication of [Formula: see text] along an ideal [Formula: see text] [Formula: see text] is studied.

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.

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).

Alpha - Skew Pi - Armendariz Rings

In this article we introduce a new concept called Alpha-skew Pi-Armendariz rings (Alpha - S Pi - AR)as a generalization of the notion of Alpha-skew Armendariz rings.Another important goal behind studying this class of rings is to employ it in order to design a modern algorithm of an identification scheme according to the evolution of using modern algebra in the applications of the field of cryptography.We investigate general properties of this concept and give examples for illustration. Furthermore, this paperstudy the relationship between this concept and some previous notions related to Alpha-skew Armendariz rings. It clearly presents that every weak Alpha-skew Armendariz ring is Alpha-skew Pi-Armendariz (Alpha-S Pi-AR). Also, thisarticle showsthat the concepts of Alpha-skew Armendariz rings and Alpha-skew Pi- Armendariz rings are equivalent in case R is 2-primal and semiprime ring.Moreover, this paper proves for a semicommutative Alpha-compatible ringR that if R[x;Alpha] is nil-Armendariz, thenR is an Alpha-S Pi-AR. In addition, if R is an Alpha - S Pi -AR, 2-primal and semiprime ring, then N(R[x;Alpha])=N(R)[x;Alpha]. Finally, we look forwardthat Alpha-skew Pi-Armendariz rings (Alpha-S Pi-AR)be more effect (due to their properties) in the field of cryptography than Pi-Armendariz rings, weak Armendariz rings and others.For these properties and characterizations of the introduced concept Alpha-S Pi-AR, we aspire to design a novel algorithm of an identification scheme.

Further results on central Armendariz rings

A ring [Formula: see text] is central Armendariz if [Formula: see text] and [Formula: see text] [Formula: see text] over [Formula: see text] satisfy [Formula: see text] then all [Formula: see text] are central. It is proved that if [Formula: see text] is a central Armendariz ring, then [Formula: see text] implies that all [Formula: see text] are in its prime radical.

?-Armendariz Rings and Related Concepts

In this paper we investigated some new properties of ?-Armendariz rings and studied the relationships between ?-Armendariz rings and central Armendariz rings, nil-Armendariz rings, semicommutative rings, skew Armendariz rings, ?-compatible rings and others. We proved that if R is a central Armendariz, then R is ?-Armendariz ring. Also we explained how skew Armendariz rings can be ?-Armendariz, for that we proved that if R is a skew Armendariz?-compatible ring, then R is ?-Armendariz. Examples are given to illustrate the relations between concepts.

Generalizations of reversible and Armendariz rings

This paper concerns several ring theoretic properties related to matrices and polynomials. The basic properties of [Formula: see text]-reversible and power-Armendariz are studied. We provide a method by which one can always construct a power-Armendariz ring but neither symmetric nor Armendariz from given any symmetric ring. We investigate next various interesting relations among ring theoretic properties containing [Formula: see text]-reversibility and power-Armendariz condition.

On Radicals of Skew Inverse Laurent Series Rings

Let R be a ring and α an automorphism of R. Amitsur proved that the Jacobson radical J(R[x]) of the polynomial ring R[x] is the polynomial ring over the nil ideal J(R[x]) ∩ R. Following Amitsur, it is shown that when R is an Armendariz ring of skew inverse Laurent series type and S is any one of the ring extensions R[x;α], R[x,x-1;α], R[[x-1;α]] and R((x-1;α)), then ℜ𝔞𝔡(S) = ℜ𝔞𝔡(R)S = Nil (S), ℜ𝔞𝔡(S) ∩ R = Nil (R), where ℜ𝔞𝔡 is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals.

On constant Armendariz rings

Let [Formula: see text] be a ring equipped with an endomorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. In this article, we study [Formula: see text]-constant Armendariz rings and radicals of the Ore extension [Formula: see text], in terms of a [Formula: see text]-constant Armendariz ring [Formula: see text] with an [Formula: see text]-condition, are determined. We prove that several properties transfer between [Formula: see text] and the Ore extension [Formula: see text], in case [Formula: see text] is [Formula: see text]-compatible [Formula: see text]-constant Armendariz.