Radical factorization in commutative rings, monoids and multiplicative lattices

Abstract

In this paper we study the concept of radical factorization in the context of abstract ideal theory in order to obtain a unified approach to the theory of factorization into radical ideals and elements in the literature of commutative rings, monoids and ideal systems. Using this approach we derive new characterizations of classes of rings whose ideals are a product of radical ideals, and we obtain also similar characterizations for classes of ideal systems in monoids and star ideals in integral domains.2000 Mathematics Subject Classification. 13A15, 13F05, 20M12, 20M13.

Products of ideals in Leavitt path algebras

“…(We note that the Bézout and hereditary properties hold for all one-sided ideals as well.) It is well-known (see, e.g., [7], [8], [9]) that these integral domains admit satisfactory factorizations of their ideals as products and intersections of special types of ideals such as prime, prime-power, primary, irreducible, semiprime, or quasi-primary. In light of the aforementioned similarities, it is natural to investigate factorizations of the ideals of a Leavitt path algebra L as products of these special types of ideals of L.…”

Radical factorization in finitary ideal systems

“…We extend the known characterizations of r-almost Dedekind r-SP-monoids for finitary ideal systems r. We consider lattices of ideals that are (a priori) neither principally generated nor modular. Thus we complement the results of [18] by describing the lattice of r-ideals in case that r is a (not necessarily modular) finitary ideal system. Let p be a modular finitary ideal system and r a finitary ideal system such that every r-ideal is a p-ideal.…”

“…Besides that, radical factorization in commutative rings with identity was investigated in [1]. Many of these results were extended in a recent paper [18] where radical factorization was studied in the context of principally generated C-lattice domains.…”

On the arithmetic of monoids of ideals

“…To mention some classical results, R is a Dedekind domain if and only if I(R) = I * (R) if and only if I(R) is a factorial monoid, and R is a Krull domain if and only if I * v (R) is a factorial monoid. Recent progress in such directions can be found in the work by Anderson, Juett, Klingler, Olberding, Reinhart, and others (e.g., [49,58,3,50,56,57,52]).…”