On some factorial properties of subrings

On properties of square-free elements in commutative cancellative monoids

“…A monoid H is called radical factorial if every element is a product of radical elements. As we already observed in [16], Lemma 3.2 b), every radical element is square-free. So we have the following diagram of relations on elements of a monoid:…”

“…Recall that every radical element is square-free ( [16], Lemma 3.2 b), so radical factorial monoids studied by Reinhart in [21] satisfy condition (i).…”

An approach to the Jacobian Conjecture in terms of irreducibility and square-freeness

“…In order to understand more general context of conditions Irr R ⊂ Sqf A and Sqf R ⊂ Sqf A when R is a subring of a domain A, we can inscribe them into the following diagram of implications ( [25], Proposition 3.3).…”

“…Its long history is full of equivalent formulations and wrong proofs. In this article we give a survey of a new purely algebraic approach to the Jacobian Conjecture in terms of irreducible elements and square-free elements, based mainly on: one of the authors' paper [22], de Bondt and Yan's paper [7], our paper [24], and our joint paper with Matysiak [25].…”

A note on square-free factorizations

“…In [3] we discuss many factorial properties of subrings, in particular involving square-free elements. The aim of this paper is to collect various ways to present an element as a product of square-free elements and to study the existence and uniqueness questions in larger classes than the class of unique factorization domains.…”