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We prove that an integral domain D is pseudo-Dedekind if and only if it satisfies a certain "Gauss's lemma for power series": c (f g) v = c (f) v c (g) v for all power series f and g over D in any (possibly infinite) number of indeterminates, where c (f) v denotes the v-closure of the content of f. We generalize this fact via semistar operations, coefficient rings with zero divisors, and fractional (semi)regular contents. Along the way, we prove several other new facts about (semistar) power series content formulas, generalizing as above on known results about Prüfer (v-multiplication) domains, (completely) integrally closed domains, and (generalized) GCD domains. [ABSTRACT FROM AUTHOR]

Several different versions of "factoriality" have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of "unique factorization rings." Our work generalizes Anderson et al.'s results about "unique factorization" in R[X], Gilmer and Parker's characterization of factorial monoid domains, and Hardy and Shores's classification of when R[S] is a principal ideal ring (for S cancellative). Along the way, we determine when R[S] is "restricted cancellative" or satisfies various "(restricted) ideal cancellation laws." [ABSTRACT FROM AUTHOR]

"Unique factorization" was central to the initial development of ideal theory. We update this topic with several new results concerning notions of "unique ideal factorization rings" with zero divisors. Along the way, we obtain new characterizations of several well-known kinds of rings in terms of their ideal factorization properties and examine when monoid rings satisfy various kinds of "unique ideal factorization." Our results include necessary and sufficient conditions for a monoid ring R [ S ] with S cancellative to be a π-ring, a higher-dimensional generalization of Hardy and Shores's classic characterization of when R [ S ] is a general Zerlegung Primideale ring. [ABSTRACT FROM AUTHOR]

We study the factorization of ideals of a commutative ring, in the context of the U-factorization framework introduced by Fletcher. This leads to several "U-factorability" properties weaker than unique U-factorization. We characterize these notions, determine the implications between them, and give several examples to illustrate the differences. For example, we show that a ring is a general ZPI-ring if and only if its monoid of ideals has unique factorization in the sense of Fletcher. We also examine how these "U-factorability" properties behave with respect to several ring-theoretic constructions. [ABSTRACT FROM AUTHOR]