Arithmetical Properties of Commutative Rings and Monoids Lecture Notes in Pure and Applied Mathematics Series

The study of nonunique factorizations of elements into irreducible elements in commutative rings and monoids has emerged as an independent area of research only over the last 30 years and has enjoyed a recent flurry of activity and advancement. This book presents the proceedings of two recent meetings that gathered key researchers from around the world to review recent major results. The first seven chapters demonstrate the diversity of approaches taken in studying nonunique factorizations and serve both as an introduction to factorization theory and as a survey of current trends and results. The remaining chapters reflect research motivated by arithmetical properties of commutative rings and monoids.

Preface 1 Non-Atomic Unique Factorization in Integral Domains 2 Divisibility Properties in Graded Integral Domains 3 Extensions of Half-Factorial Domains: A Survey 4 C-Monoids and Congruence Monoids in Krull Domains 5 Monotone Chains of Factorizations in C-Monoids 6 Transfer Principles in the Theory of Non-unique Factorizations 7 Cale Monoids, Cale Domains, and Cale Varieties 8 Weakly Krull Inside Factorial Domains 9 The m-Complement of a Multiplicative Set 10 Some Remarks on Infinite Products 11 Rings with Prime Nilradical 12 On the Ideal Generated by the Values of a Polynomial 13 Using Factorizations to Prove a Partition Identity 14 On Inside Factorial Integral Domains 15 Polynomial Separation of Points in Algebras 16 k-Factorized Elements in Telescopic Numerical Semigroups 17 Prufer Conditions in Rings with Zero-Divisors 18 Unmixedness and the Generalized Principal Ideal Theorem 19 A Note on Sets of Lengths of Powers of Elements of Finitely Generated Monoids 20 UMV-Domains 21 On Local Half-Factorial Orders 22 On Factorization in Krull Domains with Divisor Class Group Z2k 23 Integral Morphisms 24 A Special Type of Invertible Ideal 25 Factorization into Radical Ideals 26 Strongly Primary Ideals

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Scott T. Chapman, Trinity University, San Antonio, Texas, U.S.A