Factoring Ideals in Integral Domains

Abstract

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Which sets are sets of lengths in all numerical monoids?

“…Factorization Theory originated from algebraic number theory before it branched out into various subfields of algebra ( [16,11,3,2,8]). The goal is to understand from a qualitative and quantitative point of view the various phenomena of non-uniqueness of factorizations into atoms (irreducible elements) that can occur in non-factorial domains and monoids.…”

Core of ideals in integral domains

“…A domain R is said to have the trace property if for each nonzero ideal I of R, either I is invertible in R or I(R : I) is a prime ideal of R [17,18,41]. Valuation domains [ We notice that, in a domain with the trace property, every ideal I satisfies I 2 I −1 ⊆ core(I).…”

Factorization theory in commutative monoids

“…Various aspects of factorization theory could not be covered in this survey. These include factorizations in non-commutative rings and semigroups ( [86]), factorizations in commutative rings with zero-divisors ( [5]), arithmetic of non-atomic, non-BF, and non-Mori domains ( [8,23,24]), and factorizations into distinguished elements that are not irreducible (e.g., factorizations into radical ideals and others [32,81,75,76]).…”