A note on square-free factorizations

Abstract

Abstract. We analyze properties of various square-free factorizations in greatest common divisor domains (GCD-domains) and domains satisfying the ascending chain condition for principal ideals (ACCP-domains).

A Topology on Sums of Square-Free Ideals in Monoids

“…The mapping action is continuous because it is continuous over C(X, X). (5) The set of all subsets of any set X with the action of the union of sets is a monoid, but it is not a topological monoid. There is no sensible topology on the power set X that would be consistent with the operation of the sum.…”

“…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.…”

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.…”