This article explores several extensions of the Prüfer domain notion to rings with zero divisors. These extensions include semihereditary rings, rings with weak global dimension less than or equal to 1, arithmetical rings, Gaussian rings, locally Prüfer rings, strongly Prüfer rings, and Prüfer rings. The renewed interest in these properties, due to their connection to Kaplansky's Conjecture, has resulted in a large body of results shedding new light on the area. We survey the work done in this direction in the last 15 years, including results, examples and counterexamples, and a multitude of open problems.
This article surveys the known results for several related families of ring properties in the context of commutative group rings. These properties include finiteness conditions, homological conditions, and conditions that connect these two families. We briefly survey the classical results, highlight the recent progress, and point out open problems and possible future directions of investigation in these areas.
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