Abstract
In this article, we investigate the concept of radical factorization with respect to finitary ideal systems of cancellative monoids. We present new characterizations for r-almost Dedekind r-SP-monoids and provide specific descriptions of t-almost Dedekind t-SP-monoids and w-SP-monoids. We show that a monoid is a w-SP-monoid if and only if the radical of every nontrivial principal ideal is t-invertible. We characterize when the monoid ring is a w-SP-domain and describe when the *-Nagata ring is an SP-domain for a star operation * of finite type.
Highlights
The concept of factoring ideals into radical ideals has been studied by various authors
They showed that every integral domain for which every ideal is a finite product of radical ideals is an almost Dedekind domain
After that the second-named author investigated the concept of radical factorization in [19, 20] with respect to finitary ideal systems
Summary
The concept of factoring ideals into radical ideals has been studied by various authors. After that the second-named author investigated the concept of radical factorization in [19, 20] with respect to finitary ideal systems. 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. Ideal systems of monoids are a generalization of star operations of integral domains. They were studied in detail in [12]. We characterize when every principal ideal of the monoid of r-invertible r-ideals is a finite product of pairwise comparable radical principal ideals. We show that if à is a star operation of finite type of an integral domain R, the Ã-Nagata ring of R is an SP-domain if and only if R is a Ã-almost Dedekind Ã-SP-domain
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