Ring Extensions Involving Amalgamated Duplications

Abstract

Within the last decade, much attention in commutative ring theory has been drawn into the revitalized concept of an “amalgamated duplication of a ring along an ideal,” also known simply as a “bowtie ring.” The basic bowtie ring construction has roots as early as 1932, while the updated form simultaneously generalizes myriad other known constructions, including D + M rings, A + B rings, the rings A + B[X] and A + B[[X]] (for an extension ring B of the ring A), and Nagata's idealization of a module, a concept which itself has been indispensable in commutative algebra for over 50 years in providing examples of non-domains with various desired properties.