Remarks on a remark of Kaplansky
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- by William J. Heinzer and Ira J. Papick PDF
- Proc. Amer. Math. Soc. 105 (1989), 1-9 Request permission
Abstract:
In his book Commutative rings, Kaplansky makes an interesting remark following the proof of the Hilbert Basis Theorem. He says, "Justly celebrated though this proof is, it leaves one somewhat dissatisfied, since the condition that $I$ and the ${I_n}$ âs be finitely generated is by no means necessary for $J$ to be finitely generated". The purpose of our note is to elaborate on Kaplanskyâs remark.References
-
N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., 1972.
- P. Brumatti, A. Simis, and W. V. Vasconcelos, Normal Rees algebras, J. Algebra 112 (1988), no. 1, 26â48. MR 921962, DOI 10.1016/0021-8693(88)90130-5
- David E. Dobbs and Ira J. Papick, When is $D+M$ coherent?, Proc. Amer. Math. Soc. 56 (1976), 51â54. MR 409448, DOI 10.1090/S0002-9939-1976-0409448-9
- Paul Eakin and William Heinzer, Non finiteness in finite dimensional Krull domains, J. Algebra 14 (1970), 333â340. MR 254023, DOI 10.1016/0021-8693(70)90109-2
- Robert Gilmer, Multiplicative ideal theory, Pure and Applied Mathematics, No. 12, Marcel Dekker, Inc., New York, 1972. MR 0427289
- Robert Gilmer and Raymond C. Heitmann, On $\textrm {Pic}(R[X])$ for $R$ seminormal, J. Pure Appl. Algebra 16 (1980), no. 3, 251â257. MR 558489, DOI 10.1016/0022-4049(80)90030-4
- Sarah Glaz and Wolmer V. Vasconcelos, Flat ideals. III, Comm. Algebra 12 (1984), no. 1-2, 199â227. MR 732194, DOI 10.1080/00927878408822998
- Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivitĂŠ. Techniques de âplatificationâ dâun module, Invent. Math. 13 (1971), 1â89 (French). MR 308104, DOI 10.1007/BF01390094
- Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
- Hwa Tsang Tang, Gaussâ lemma, Proc. Amer. Math. Soc. 35 (1972), 372â376. MR 302638, DOI 10.1090/S0002-9939-1972-0302638-1
- Wolmer V. Vasconcelos, The rings of dimension two, Lecture Notes in Pure and Applied Mathematics, Vol. 22, Marcel Dekker, Inc., New York-Basel, 1976. MR 0427290
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 105 (1989), 1-9
- MSC: Primary 13A15
- DOI: https://doi.org/10.1090/S0002-9939-1989-0973834-8
- MathSciNet review: 973834