## Commutative Semigroup RingsCommutative Semigroup Rings was the first exposition of the basic properties of semigroup rings. Gilmer concentrates on the interplay between semigroups and rings, thereby illuminating both of these important concepts in modern algebra. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

COMMUTATIVE SEMIGROUPS | 1 |

Basic Notions | 2 |

Cyclic Semigroups and Numerical Monoids | 9 |

Ordered Semigroups | 22 |

Congruences | 28 |

Noetherian Semigroups | 39 |

Factorization in Commutative Monoids | 52 |

SEMIGROUP RINGS AND THEIR DISTINGUISHED ELEMENTS | 63 |

The Divisor Class Group of a Krull Monoid Domain | 208 |

RINGTHEORETIC PROPERTIES OF MONOID RINGS | 225 |

Monoid Rings as von Neumann Regular Rings | 226 |

Monoid Rings as Priifer Rings | 240 |

Monoid Rings as Arithmetical Rings the Case Where S is not TorsionFree | 253 |

Chain Conditions in Monoid Rings | 271 |

DIMENSION THEORY AND THE ISOMORPHISM PROBLEMS | 287 |

Dimension Theory of Monoid Rings | 288 |

Semigroup Rings | 64 |

Zero Divisors | 82 |

Nilpotent Elements | 97 |

Idempotents | 113 |

Units | 129 |

RINGTHEORETIC PROPERTIES OF MONOID DOMAINS | 145 |

Integral Dependence for Monoid Rings | 147 |

Monoid Domains as Prufer Domains | 162 |

Monoid Domains as Factorial Domains | 171 |

Monoid Domains as Krull Domains | 190 |

Rautomorphisms of RS | 303 |

Coefficient Rings in Isomorphic Monoid Rings I | 317 |

Coefficient Rings in Isomorphic Monoid Rings II | 337 |

Monoids in Isomorphic Monoid Rings a Survey | 351 |

SELECTED BIBLIOGRAPHY | 355 |

363 | |

INDEX OF MAIN NOTATION | 371 |

INDEX OF SOME MAIN RESULTS | 375 |

### Common terms and phrases

additive algebraic arithmetical asserted Assume called cancellative cancellative monoid canonical Chapter characteristic Choose clear common completes the proof concerning conclude congruence Consequently consider contains converse Corollary cyclic decomposition Dedekind domain defined definition denote determining direct sum equality equivalent example exists extension fact factorial field finite follows given group G group rings hence homomorphism idempotent implies inclusion induction integral domain integrally closed invertible isomorphism Krull domain locally mapping Math maximal ideal minimal monoid ring Moreover multiplicative nilpotent nilradical Noetherian nonzero element notation periodic Priifer primary prime ideal problem proof of Theorem properties prove quotient group reduced regular relation Remarks result satisfies Section semigroup rings shows statement submonoid subring subset suffices theory torsion—free unique unit unitary ring valuation write zero divisor