Bounded factorization and the ascending chain condition on principal ideals in generalized power series rings. (English) Zbl 1530.13034
Chabert, Jean-Luc (ed.) et al., Algebraic, number theoretic, and topological aspects of ring theory. Selected papers based on the cancelled conference on rings and polynomials, July 2020, and the fourth international meeting on integer-valued polynomials and related topics, CIRM, Luminy, France, July 19–24, 2021. Cham: Springer. 135-153 (2023).
Summary: We determine necessary and sufficient conditions for broad classes of generalized power series rings to satisfy the ascending chain condition on principal ideals or possess the bounded factorization property. Along the way, we consider when a generalized power series ring is domainlike or (weakly) présimplifiable. As corollaries to our general theorems, we derive new factorization-theoretic results about (Laurent) power series rings and the “large polynomial rings” of Halter-Koch.
For the entire collection see [Zbl 1515.13002].
For the entire collection see [Zbl 1515.13002].
MSC:
| 13F15 | Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) |
| 13A05 | Divisibility and factorizations in commutative rings |
| 13F25 | Formal power series rings |
| 20M25 | Semigroup rings, multiplicative semigroups of rings |
Keywords:
bounded factorization ring; ascending chain condition on principal ideals; generalized power series ring; (Weakly) présimplifiable ring; domainlike ringReferences:
| [1] | D.D. Anderson and O.A. Al-Mallah. Commutative group rings that are présimplifiable or domainlike. J. Algebra Appl., 16:1750019, 2017. · Zbl 1355.13002 · doi:10.1142/S0219498817500190 |
| [2] | D.D. Anderson, D.F. Anderson, and M. Zafrullah. Factorization in integral domains. J. Pure Appl. Algebra, 69:1-19, 1990. · Zbl 0727.13007 · doi:10.1016/0022-4049(90)90074-R |
| [3] | D.D. Anderson and S. Chun. Associate elements in commutative rings. Rocky Mountain J. Math., 44:717-731, 2014. · Zbl 1302.13001 · doi:10.1216/RMJ-2014-44-3-717 |
| [4] | D.D. Anderson and A. Ganatra. Bounded factorization rings. Comm. Algebra, 35:3892-3903, 2007. · Zbl 1159.13001 · doi:10.1080/00927870701509230 |
| [5] | D.D. Anderson and J.R. Juett. Long length functions. J. Algebra, 426:327-343, 2015. · Zbl 1314.13005 · doi:10.1016/j.jalgebra.2014.12.016 |
| [6] | D.D. Anderson and J.R. Juett. Length functions in commutative rings with zero divisors. Comm. Algebra, 45:1584-1600, 2017. · Zbl 1369.13001 · doi:10.1080/00927872.2016.1222400 |
| [7] | D.D. Anderson, J.R. Juett, and C.P. Mooney. Factorization of ideals. Comm. Algebra, 47:1742-1772, 2019. · Zbl 1431.13003 · doi:10.1080/00927872.2018.1517359 |
| [8] | D.D. Anderson and S. Valdes-Leon. Factorization in commutative rings with zero divisors. Rocky Mountain J. Math., 26:439-480, 1996. · Zbl 0865.13001 · doi:10.1216/rmjm/1181072068 |
| [9] | D.D. Anderson and S. Valdes-Leon. Factorization in commutative rings with zero divisors, II. In D.D. Anderson, editor, Factorization in integral domains, pages 197-219. Marcel Dekker, 1997. · Zbl 0878.13001 |
| [10] | D.F. Anderson and F. Gotti. Bounded and finite factorization domains. In A. Badawi and J. Coykendall, editors, Rings, monoids, and module theory. Springer-Verlag, Singapore, 2022. |
| [11] | M. Axtell, S.J. Forman, and J. Stickles. Properties of domainlike rings. Tamkang J. Math., 40:151-164, 2009. · Zbl 1188.13001 · doi:10.5556/j.tkjm.40.2009.464 |
| [12] | N.P. Aylesworth and J.R. Juett. Generalized power series with a limited number of factorizations. J. Commut. Algebra, 14:471-507, 2022. · Zbl 1506.13003 · doi:10.1216/jca.2022.14.471 |
| [13] | R. Baer. Radical ideals. Amer. J. Math., 65:537-568, 1943. · Zbl 0060.07104 · doi:10.2307/2371865 |
| [14] | A. Benhissi and M. Eljeri. MC-extensions: examples, zero-divisors graph, and colorability. Comm. Algebra, 40:104-124, 2012. · Zbl 1267.13003 · doi:10.1080/00927872.2010.524904 |
| [15] | G.M. Benkart, A.M. Gaglione, W.D. Joyner, M.E. Kidwell, M.D. Meyerson, D. Spellman, and W.P. Wardlaw. Principal ideals and associate rings. JP J. Algebra Number Theory Appl., 2:181-193, 2002. · Zbl 1046.13004 |
| [16] | A. Bouvier. Anneaux présimplifiables et anneaux atomiques. C.R. Acad. Sci. Paris, 272:992-994, 1971. · Zbl 0216.32804 |
| [17] | A. Bouvier. Anneaux présimplifiables. C.R. Acad. Sci. Paris, 274:A1605-A1607, 1972. · Zbl 0244.13009 |
| [18] | A. Bouvier. Résultats nouveaux sur les anneaux présimplifiables. C.R. Acad. Sci. Paris, 275:A955-A957, 1972. · Zbl 0242.13002 |
| [19] | A. Bouvier. Anneaux présimplifiables. Rev. Roumaine Math. Pures Appl., 19:713-724, 1974. · Zbl 0289.13010 |
| [20] | E.D. Cashwell and C.J. Everett. Formal power series. Pacific J. Math., 13:45-64, 1963. · Zbl 0117.02603 · doi:10.2140/pjm.1963.13.45 |
| [21] | S.T. Chapman, F. Gotti, and M. Gotti. Factorization invariants of Puiseux monoids generated by geometric sequences. Comm. Algebra, 48:380-396, 2020. · Zbl 1442.20036 · doi:10.1080/00927872.2019.1646269 |
| [22] | C.C. Cheng, J.H. McKay, J. Towber, S.S. Wang, and D.L. Wright. Reversion of power series and the extended Raney coefficients. Trans. Amer. Math. Soc., 349:1769-1782, 1997. · Zbl 0868.13019 · doi:10.1090/S0002-9947-97-01781-9 |
| [23] | R.A.C. Edmonds and J.R. Juett. Associates, irreducibility, and bounded factorization in monoid rings with zero divisors. Comm. Algebra, 49:1836-1860, 2021. · Zbl 1461.13026 · doi:10.1080/00927872.2020.1857391 |
| [24] | G.A. Elliott and P. Ribenboim. Fields of generalized power series. Arch. Math., 54:365-371, 1990. · Zbl 0676.13010 · doi:10.1007/BF01189583 |
| [25] | D. Frohn. A counterexample concerning ACCP in power series rings. Comm. Algebra, 30:2961-2966, 2002. · Zbl 1008.13005 · doi:10.1081/AGB-120004001 |
| [26] | D. Frohn. Modules with \(n\)-acc and the acc on certain types of annihilators. J. Algebra, 256:467-483, 2002. · Zbl 1047.13004 · doi:10.1016/S0021-8693(02)00039-X |
| [27] | D. Frohn. ACCP rises to the polynomial ring if the ring has only finitely many associated primes. Comm. Algebra, 32:1213-1218, 2004. · Zbl 1094.13029 · doi:10.1081/AGB-120027975 |
| [28] | A. Geroldinger and F. Halter-Koch. Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory, volume 278 of Pure and Applied Mathematics (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2006. · Zbl 1113.11002 |
| [29] | R. Gilmer. Multiplicative Ideal Theory, Queen’s Papers in Pure and Appl. Math., vol. 90. Queen’s University, Kingston, Ontario, 1992. · Zbl 0804.13001 |
| [30] | R. Gilmer and T. Parker. Divisibility properties in semigroup rings. Michigan Math. J., 21:65-86, 1974. · Zbl 0285.13007 · doi:10.1307/mmj/1029001210 |
| [31] | F. Gotti. Systems of sets of lengths of Puiseux monoids. J. Pure Appl. Algebra, 223:1856-1868, 2019. · Zbl 1459.20045 · doi:10.1016/j.jpaa.2018.08.004 |
| [32] | F. Halter-Koch. On the algebraic and arithmetical structure of generalized polynomial algebras. Rend. Semin. Mat. Univ. Padova, 90:121-140, 1993. · Zbl 0795.13008 |
| [33] | W.J. Heinzer and D.C. Lantz. Commutative rings with ACC on \(n\)-generated ideals. J. Algebra, 80:261-278, 1983. · Zbl 0512.13011 · doi:10.1016/0021-8693(83)90031-5 |
| [34] | W.J. Heinzer and D.C. Lantz. ACCP in polynomial rings: a counterexample. Proc. Amer. Math. Soc., 121:975-977, 1994. · Zbl 0828.13013 · doi:10.1090/S0002-9939-1994-1232140-9 |
| [35] | H. Kim. Factorization in monoid domains. Comm. Algebra, 29:1853-1869, 2001. · Zbl 1009.13005 · doi:10.1081/AGB-100002153 |
| [36] | P. Ribenboim. Generalized power series rings. In Lattices, semigroups, and universal algebra (Lisbon, 1988), pages 271-277. Plenum, New York, 1990. · Zbl 0757.06009 |
| [37] | P. Ribenboim. Rings of generalized power series: nilpotent elements. Abh. Math. Sem. Univ. Hamburg, 61:15-33, 1991. · Zbl 0751.13005 · doi:10.1007/BF02950748 |
| [38] | P. Ribenboim. Noetherian rings of generalized power series. J. Pure Appl. Algebra, 79:293-312, 1992. · Zbl 0761.13007 · doi:10.1016/0022-4049(92)90056-L |
| [39] | P. Ribenboim. Rings of generalized power series II: Units and zero-divisors. J. Algebra, 168:71-89, 1994. · Zbl 0806.13011 · doi:10.1006/jabr.1994.1221 |
| [40] | P. Ribenboim. Special properties of rings of generalized power series. J. Algebra, 173:566-586, 1995. · Zbl 0852.13008 · doi:10.1006/jabr.1995.1103 |
| [41] | P. Ribenboim. Semisimple rings and von Neumann regular rings of generalized power series. J. Algebra, 198:327-338, 1997. · Zbl 0890.16004 · doi:10.1006/jabr.1997.7063 |
| [42] | G. Xin. A fast algorithm for MacMahon’s partition analysis. Electron. J. Combin., 11, 2004. · Zbl 1066.11060 |
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