On properties of square-free elements in commutative cancellative monoids

Abstract

We discuss various square-free factorizations in monoids in the context of: atomicity, ascending chain condition for principal ideals, decomposition, and a greatest common divisor property. Moreover, we obtain a full characterization of submonoids of factorial monoids in which all square-free elements of a submonoid are square-free in a monoid. We also present factorial properties implying that all atoms of a submonoid are square-free in a monoid.(1.1) A(M) ⊂ S(H).

On properties of composites and monoid domains

“…More information about Schreier and pre-Schreier domains we can see in many works, e.g. in [1], [5], [7], [14], [18], respectively. Lemma 2.13.…”

“…In [18] Proposition 8.6 with Jȩdrzejewicz, Marciniak and Zieliński we received the characterization of square-free elements in composite A+XB[X], for A ⊂ B fields, i.e. Proposition 2.16.…”

“…That is why I decided to examine all possible properties of composites for commutative algebra. I put the first results in [18], and the next results I put in this article.…”

On Certain Properties of Square-Free Numbers and the Function <em>s(n)</em>

“…In section 4 we provide an encouraging overview of the initial theory of square-free factorizations for commutative cancellative monoids (in particular for Z), which was pioneered in the papers [4], [3], [6] and [7]. Let us recall (according to [2]) that by a commutative cancellative monoid we mean the semigroup H satisfying the law of contraction, i.e.…”

A Topology on Sums of Square-Free Ideals in Monoids

“…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.…”