SP-rings with zero-divisors

Abstract

We characterize the commutative rings whose ideals (resp. regular ideals) are products of radical ideals

Radical factorization in finitary ideal systems

“…Clearly, the prime r M -ideals of H M form a chain and every nontrivial prime r M -ideal of H M contains an r M -invertible radical r M -ideal of H M . Therefore, H M is a DVM by (1), and hence H is an r-almost Dedekind monoid. It follows from [19,Corollary 3.4] that H is an r-SP-monoid.…”

“…Then (JL) rp ∈ I * rp (H) ⊆ I r (H). We infer that (JL) rp = ((JL) rp ) r = (JL) r , since r p ≤ r by (1).…”

“…Further progress in describing SP-domains was made in [9,13,15]. 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.…”

“…This implies that I = ( n i=1 I i ) r . Let i ∈ [1, n].It follows by Lemma 5.1(2) that I i is a radical r-ideal of H. Furthermore, if i ∈ [1, n − 1], then I i ⊆ I i+1 by Lemma 5.1(1).…”

“…(4) A = J[Z] for some t-ideal J of R and some t-ideal Z of H if and only if A is a homogeneous t-ideal of R[H] if and only if A = F t for some nonempty set F of nonzero homogeneous elements of R[H]. (5) If R[H] is integrally closed and A is a t-ideal of R[H] that contains a nonzero homogeneous element of R[H], then A is homogeneous.Proof (1). Recall that there is some total order ≤ on H that is compatible with the monoid operation on H.…”