Ore sets, denominator sets and the left regular left quotient ring of a ring

Abstract

The aim of the papers is to describe the left regular left quotient ring [Formula: see text] and the right regular right quotient ring [Formula: see text] for the following algebras [Formula: see text]: [Formula: see text] is the algebra of one-sided inverses, where [Formula: see text], [Formula: see text] is the algebra of scalar integro-differential operators and the Jacobian algebra [Formula: see text]. The sets of left and right regular elements of the algebras [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are described. A progress is made on the following conjecture, [12]: [Formula: see text] is the algebra of polynomial integro-differential operators and [Formula: see text] is the classical quotient ring (of fractions) of the [Formula: see text]’th Weyl algebra [Formula: see text], i.e. a criterion is given when the isomorphism holds. We produce several general constructions of left Ore and left denominator sets that appear naturally in applications and are of independent interest and use them to produce explicit left denominator sets that give the localization ring isomorphic to [Formula: see text] or [Formula: see text] or [Formula: see text] where [Formula: see text]. Several characterizations of one-sided regular elements of a ring are given in module-theoretic and one-sided-ideal-theoretic way.

The authors' abstract, as published at the source. Journal of Algebra and Its Applications, 2026 · DOI ↗