## Multiplicative Ideal Theory and Factorization Theory: Commutative and Non-commutative PerspectivesScott Chapman, Marco Fontana, Alfred Geroldinger, Bruce Olberding This book consists of both expository and research articles solicited from speakers at the conference entitled "Arithmetic and Ideal Theory of Rings and Semigroups," held September 22–26, 2014 at the University of Graz, Graz, Austria. It reflects recent trends in multiplicative ideal theory and factorization theory, and brings together for the first time in one volume both commutative and non-commutative perspectives on these areas, which have their roots in number theory, commutative algebra, and algebraic geometry. Topics discussed include topological aspects in ring theory, Prüfer domains of integer-valued polynomials and their monadic submonoids, and semigroup algebras. It will be of interest to practitioners of mathematics and computer science, and researchers in multiplicative ideal theory, factorization theory, number theory, and algebraic geometry. |

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### Contents

1 | |

About Number Fields with Pólya Group of Order leq2 | 23 |

The Interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics | 43 |

Ring and Semigroup Constructions | 96 |

A Survey | 117 |

Relative Polynomial Closure and Monadically Krull Monoids of IntegerValued Polynomials | 144 |

An Overview of the Computational Aspects of Nonunique Factorization Invariants | 159 |

The Global Case | 182 |

Lobal Properties of Integral Domains | 233 |

Noetherian Semigroup Algebras and Beyond | 255 |

Topological Aspects of Irredundant Intersections of Ideals and Valuation Rings | 277 |

Idempotent Pairs and PRINC Domains | 308 |

Some Recent Results and Open Problems on Sets of Lengths of Krull Monoids with Finite Class Group | 323 |

A Survey | 353 |

403 | |

Prüfer Domains of IntegerValued Polynomials | 219 |