Abstract
“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 with S cancellative to be a π-ring, a higher-dimensional generalization of Hardy and Shores’s classic characterization of when is a general Zerlegung Primideale ring.
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