On some factorial properties of subrings

Abstract

We discuss various factorial properties of subrings as well as properties involving irreducible and square-free elements, in particular ones connected with Jacobian conditions.

A Topology on Sums of Square-Free Ideals in Monoids

“…For example, if X = {a, b, c}, then there is no way to define the neighborhoods of {a, b} such that they are closed to the sum. (6) The set of all strings over any alphabet with the operation of concatenation is a monoid, but is not a topological monoid. There is no natural metric or norm on the set of strings that would be consistent with concatenation.…”

“…Another motivation is the article [10], where the properties of square-free ideals are described. Recall that the ideal I of a ring R is called square-free if for every x ∈ R, if x 2 ∈ I, then x ∈ I. Square-free ideals are a consequence of research on the theory of square-free factorizations, the results of which can be found in the papers [4][5][6][7]9,11] (in the case of radical factorizations) and in the author's doctoral thesis, which was highly appreciated by Professor Tadeusz Krasi ński from the University of Łódź in a review, motivating the author to further work on square-free factorizations.…”

The Jacobian Conjecture and Idempotent Ideals

“…It is worth noting that in [4] the authors showed the relationship between the Jacobian hypothesis and irreducible and square-free elements in certain rings of polynomials. In this article, we will also show relationships, although not motivated by [4].…”