A two-dimensional non-Noetherian factorial ring
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- by Robert Gilmer PDF
- Proc. Amer. Math. Soc. 44 (1974), 25-30 Request permission
Abstract:
Let $R$ be a commutative ring with identity and let $G$ be an abelian group of torsion-free rank $\alpha$. If $\{ {X_\lambda }\}$ is a set of indeterminates over $R$ of cardinality $\alpha$, then the group ring of $G$ over $R$ and the polynomial ring $R[\{ {X_\lambda }\} ]$ have the same (Krull) dimension. The preceding result and a theorem due to T. Parker and the author imply that for each integer $k \geqq 2$, there is a $k$-dimensional non-Noetherian unique factorization domain of arbitrary characteristic.References
- Jimmy T. Arnold and Robert Gilmer, Dimension sequences for commutative rings, Bull. Amer. Math. Soc. 79 (1973), 407–409. MR 313244, DOI 10.1090/S0002-9904-1973-13188-X
- Jimmy T. Arnold and Robert Gilmer, The dimension sequence of a commutative ring, Amer. J. Math. 96 (1974), 385–408. MR 364221, DOI 10.2307/2373549
- Ian G. Connell, On the group ring, Canadian J. Math. 15 (1963), 650–685. MR 153705, DOI 10.4153/CJM-1963-067-0 John E. David, Some non-Noetherian factorial rings, Dissertation, University of Rochester, Rochester, N.Y., 1972.
- John David, A non-Noetherian factorial ring, Trans. Amer. Math. Soc. 169 (1972), 495–502. MR 308114, DOI 10.1090/S0002-9947-1972-0308114-9
- L. Fuchs, Abelian groups, International Series of Monographs on Pure and Applied Mathematics, Pergamon Press, New York-Oxford-London-Paris, 1960. MR 0111783
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
- Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
- Robert Gilmer, Multiplicative ideal theory, Pure and Applied Mathematics, No. 12, Marcel Dekker, Inc., New York, 1972. MR 0427289
- Robert Gilmer and Tom Parker, Divisibility properties in semigroup rings, Michigan Math. J. 21 (1974), 65–86. MR 342635
- Robert Gordon and J. C. Robson, Krull dimension, Memoirs of the American Mathematical Society, No. 133, American Mathematical Society, Providence, R.I., 1973. MR 0352177
- Nathan Jacobson, Lectures in Abstract Algebra. Vol. I. Basic Concepts, D. Van Nostrand Co., Inc., Toronto-New York-London, 1951. MR 0041102
- Joachim Lambek, Lectures on rings and modules, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. With an appendix by Ian G. Connell. MR 0206032
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge University Press, London, 1968. MR 0231816
- L. Pontrjagin, The theory of topological commutative groups, Ann. of Math. (2) 35 (1934), no. 2, 361–388. MR 1503168, DOI 10.2307/1968438
- Rudolf Rentschler and Pierre Gabriel, Sur la dimension des anneaux et ensembles ordonnés, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A712–A715 (French). MR 224644
- Joseph J. Rotman, The theory of groups. An introduction, Allyn and Bacon, Inc., Boston, Mass., 1965. MR 0204499
- P. F. Smith, On the dimension of group rings, Proc. London Math. Soc. (3) 25 (1972), 288–302. MR 314952, DOI 10.1112/plms/s3-25.2.288
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 44 (1974), 25-30
- MSC: Primary 13F15
- DOI: https://doi.org/10.1090/S0002-9939-1974-0335500-0
- MathSciNet review: 0335500