A ring has bounded factorizations if every cancellative nonunit a∈R can be written as a product of atoms and there is a bound λ(a) on the lengths of such factorizations. The bounded factorization ...property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations.
Let R be a unitary prime ring with the maximal left ring of quotients
. We completely characterize the forms of additive maps
such that
whenever
satisfy xy = 0 = yx under the assumptions that R ...contains a nontrivial idempotent and characteristic of R is not 2. This partially generalizes a result of T.K. Lee in Bi-additive maps of ζ-Lie product type vanishing on zero products of XY and Y X. Comm Algebra. 2017;45(8):3449-3467.
In the present paper, we prove some commutativity theorems for a prime ring with involution in which generalized derivations satisfy certain differential identities. Some well known results on ...commutativity of prime rings have been obtained. Also, we provide an example to show that the assumed restriction imposed on the involution is not superfluous.
If H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of ...sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.
Let $R$ be a prime ring and $L$ a non-central Lie ideal of $R.$ The purpose of this paper is to describe generalized derivations of $R$ satisfying some algebraic identities locally on $L.$ More ...precisely, we consider two generalized derivations $F_1$ and $F_2$ of a prime ring $R$ satisfying one of the following identities:1. $F_1(x)\circ y +x \circ F_2(y) =0,$2. $F_1(x),y + F_2(x,y) =0,$for all $x,y$ in a non-central Lie ideal $L$ of $R.$ Furthermore, as an application, we study continuous generalized derivations satisfying similar algebraic identities with power values on nonvoidopen subsets of a prime Banach algebra $A$. Our topological approach is based on Baire'scategory theorem and some properties from functional analysis.
Let R be a non-commutative prime ring of characteristic different from 2, Q
r
its right Martindale quotient ring and C its extended centroid. Suppose that L is a non-central Lie ideal of R, F a ...non-zero generalized skew derivation of R and
a fixed integer. If
for all
then one of the following holds:
R satisfies
the standard polynomial identity on four non-commuting variables;
there exists
such that
for all
with
Abstract
Let be a prime ring and a non-central Lie ideal of .
In this paper, we aim to classify the generalized derivations of satisfying some algebraic identities with power values on .
Moreover, ...the same identities are studied locally on a two nonvoid open subsets of a prime Banach algebra.
Transfer Krull monoids are a recently introduced class of monoids and include the multiplicative monoids of all commutative Krull domains as well as of wide classes of non-commutative Dedekind ...domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between 1 and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.
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