Decompositions of commutative monoid congruences and binomial ideals
Thomas Kahle, Ezra Miller · Algebra & Number Theory · 2014
Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings.These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects.The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras.The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground field hypotheses.Binomial primary decompositions are easily recovered from mesoprimary decomposition. 1. Introduction 1298 2. Taxonomy of congruences on monoids 1305 3.Primary decomposition and localization in monoids 1310 4. Witnesses and associated prime ideals of congruences 1313 5. Associated prime congruences 1319 6. Characterization of mesoprimary congruences 1320 7. Coprincipal congruences 1322 8. Mesoprimary decompositions of congruences 1326 9. Augmentation ideals, kernels, and nils 1329 10.Taxonomy of binomial ideals in monoid algebras 1332 11.Monomial localization, characters, and mesoprimes 1337 12. Coprincipal and mesoprimary components of binomial ideals1340 13.Mesoprimary decomposition of binomial ideals 1346 14.Binomial localization 1347 15.Primary decomposition of binomial ideals 1349 16.