General w-ZPI-Rings and a Tool for Characterizing Certain Classes of Monoid Rings

Abstract The w-operation, which is in some respects a “better behaved” variant of the classic t-operation, has recently been an object of intense study. In this article, we introduce and study general w-ZPI-rings, which are commutative rings where every proper w-ideal is a w-product of prime w-ideals. We give several characterizations of general w-ZPI-rings and investigate when a monoid ring with S cancellative is a general w-ZPI-ring. On the way to answering the latter question, we formulate a reusable tool for reducing certain monoid ring classification problems to the monoid domain special case.

[1]  G. Chang,et al.  Prime Factorization of ideals in commutative rings, with a focus on Krull rings , 2022, 2204.12098.

[2]  Jun Seok Oh,et al.  The monoid of regular elements in commutative rings with zero divisors , 2021, Communications in Algebra.

[3]  J. R. Juett,et al.  Unique factorization of ideals in commutative rings with zero divisors , 2021 .

[4]  J. R. Juett,et al.  Unique factorization properties in commutative monoid rings with zero divisors , 2021, Semigroup Forum.

[5]  J. Elliott.  Rings, Modules, and Closure Operations , 2019, Springer Monographs in Mathematics.

[6]  G. Angermüller.  Strong atoms in Krull monoids , 2019, Semigroup Forum.

[7]  A. Geroldinger,et al.  Sets of arithmetical invariants in transfer Krull monoids , 2018, Journal of Pure and Applied Algebra.

[8]  J. R. Juett,et al.  U-factorization of ideals , 2018, Communications in Algebra.

[9]  C. Jayaram.  Weak π-rings , 2017 .

[10]  Hwankoo Kim,et al.  Foundations of Commutative Rings and Their Modules , 2017 .

[11]  D. D. Anderson,et al.  Commutative group rings that are présimplifiable or domainlike , 2017 .

[12]  Hwankoo Kim,et al.  w-Modules over Commutative Rings , 2016 .

[13]  Marina Daecher,et al.  Introduction To Cyclotomic Fields , 2016 .

[14]  Lei Qiao,et al.  THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS , 2015 .

[15]  Hwankoo Kim,et al.  Two generalizations of projective modules and their applications , 2015 .

[16]  Sebastian Ramacher,et al.  On v-Marot Mori rings and C-rings , 2014, 1401.2761.

[17]  J. R. Juett.  Generalized comaximal factorization of ideals , 2012 .

[18]  Iames A. HUCKABAi.  THE INTEGRAL CLOSURE OF A NOETHERIAN RING BY , 2010 .

[19]  Hwankoo Kim,et al.  Some Characterizations of Krull Monoids , 2007 .

[20]  Paul Garrett,et al.  Commutative rings I , 2007 .

[21]  M. Fontana,et al.  A generalization of Kronecker function rings and Nagata rings , 2006, math/0604040.

[22]  Alfred Geroldinger,et al.  Non-Unique Factorizations : Algebraic, Combinatorial and Analytic Theory , 2006 .

[23]  R. G. Swan,et al.  Unique comaximal factorization , 2004 .

[24]  A. Okabe.  On generalized Dedekind domains , 2004 .

[25]  Floris Ernst,et al.  Multiplicative Ideal Theory , 2004 .

[26]  Marco Fontana,et al.  Nagata Rings, Kronecker Function Rings, and Related Semistar Operations , 2003, math/0302163.

[27]  S. McAdam.  UNIQUE FACTORIZATION OF MONIC POLYNOMIALS , 2001 .

[28]  Nicholas Roersma.  U-factorizations in Commutative Rings with Zero Divisors , 2001 .

[29]  M. Zafrullah.  Putting T-Invertibility to Use , 2000 .

[30]  R. McCasland,et al.  On Strong Mori domains , 1999 .

[31]  D. D. Anderson,et al.  Unique factorization rings with zero divisors , 1999 .

[32]  S. M. Bhatwadekar.  Projective Modules Over Polynomial Rings , 1999 .

[33]  Franz Halter-Koch,et al.  Ideal Systems: An Introduction to Multiplicative Ideal Theory , 1998 .

[34]  R. McCasland,et al.  On w-modules over strong mori domains , 1997 .

[35]  D. D. Anderson,et al.  Factorization in Commutative Rings with Zero Divisors, III , 1996 .

[36]  Charles Ching-An Cheng,et al.  Resultants of cyclotomic polynomials , 1995 .

[37]  F. Halter-Koch.  A characterization of Krull rings with zero divisors , 1993 .

[38]  B. Kang.  Characterizations of Krull rings with zero divisors , 1991 .

[39]  Geroldinger Alfred.  On the arithmetic of certain not integrally closed noetheian integral domains , 1991 .

[40]  B. Kang.  On the converse of a well-known fact about Krull domains , 1989 .

[41]  Miles Reid,et al.  Commutative Ring Theory , 1989 .

[42]  B. Kang.  Prüfer v-multiplication domains and the ring R[X]Nv , 1989 .

[43]  J. Huckaba.  Commutative Rings with Zero Divisors , 1988 .

[44]  David F. Anderson,et al.  The rings R(X) and R〈X〉 , 1985 .

[45]  R. Gilmer,et al.  Commutative Semigroup Rings , 1984 .

[46]  Huah Chu.  On the of graded rings , 1984 .

[47]  A. Bouvier.  The Local Class Group of a Krull Domain , 1983, Canadian Mathematical Bulletin.

[48]  David F. Anderson,et al.  Divisibility Properties of Graded Domains , 1982, Canadian Journal of Mathematics.

[49]  D. D. Anderson,et al.  Globalization of some local properties in Krull domains , 1982 .

[50]  E. Jespers,et al.  Ω-krull rings. I , 1982 .

[51]  D. Robinson.  A Course in the Theory of Groups , 1982 .

[52]  Leo G. Chouinard,et al.  Krull semigroups and divisor class groups , 1981 .

[53]  T. Shores,et al.  Arithmetical Semigroup Rings , 1980, Canadian Journal of Mathematics.

[54]  E. Houston,et al.  Some remarks on star-operations , 1980 .

[55]  D. D. Anderson.  π-domains, overrings, and divisorial ideals , 1978, Glasgow Mathematical Journal.

[56]  D. D. Anderson,et al.  Some Remarks on the Ring R(X) , 1978 .

[57]  W. Vasconcelos,et al.  Flat ideals II , 1977 .

[58]  D. D. Anderson.  Abstract commutative ideal theory without chain condition , 1976 .

[59]  Warren D. Nichols,et al.  The Krull intersection theorem. II. , 1976 .

[60]  D. D. Anderson,et al.  Multiplication Ideals, Multiplication Rings, and the Ring R(X) , 1976, Canadian Journal of Mathematics.

[61]  Thomas H. Parker,et al.  Divisibility properties in semigroup rings. , 1974 .

[62]  Thomas H. Parker,et al.  Semigroup rings as Prüfer rings , 1974 .

[63]  K. Levitz.  A characterization of general Z.P.I.-rings. II , 1972 .

[64]  K. Levitz.  A Characterization of General Z.P.I.-Rings , 1972 .

[65]  P. Samuel,et al.  About Euclidean rings , 1971 .

[66]  E. Evans.  Zero divisors in Noetherian-like rings , 1971 .

[67]  C. Fletcher.  The structure of unique factorization rings , 1970, Mathematical Proceedings of the Cambridge Philosophical Society.

[68]  C. Fletcher.  Unique factorization rings , 1969, Mathematical Proceedings of the Cambridge Philosophical Society.

[69]  D. Northcott,et al.  Lessons on rings, modules and multiplicities , 1968 .

[70]  H. Butts,et al.  Almost Multiplication Rings , 1965, Canadian Journal of Mathematics.

[71]  S. Mori.  Allgemeine Z.P.I.-Ringe , 1940 .

[72]  S. Mori.  Über die Produktzerlegung der Hauptideale. III , 1938 .

[73]  W. Krull,et al.  Allgemeine Bewertungstheorie. , 1932 .

[74]  Richard Dedekind,et al.  Sur la théorie des nombres entiers algébriques , 1877 .

[75]  E. Kummer.  Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren. , 1847 .