On the embedded primary components of ideals. IV

Authors:
William Heinzer, L. J. Ratliff and Kishor Shah

Journal:
Trans. Amer. Math. Soc. **347** (1995), 701-708

MSC:
Primary 13E05; Secondary 13H99

MathSciNet review:
1249882

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Abstract: The results in this paper expand the fundamental decomposition theory of ideals pioneered by Emmy Noether. Specifically, let be an ideal in a local ring that has as an embedded prime divisor, and for a prime divisor of let be the set of irreducible components of that are -primary (so there exists a decomposition of as an irredundant finite intersection of irreducible ideals that has as a factor). Then the main results show: (a) ( is a MEC of in case is maximal in the set of -primary components of ); (b) if is an irredundant irreducible decomposition of such that is -primary if and only if , then is an irredundant irreducible decomposition of a MEC of , and, conversely, if is a MEC of and if (resp., ) is an irredundant irreducible decomposition of (resp., ) such that are the -primary ideals in , then and is an irredundant irreducible decomposition of ; (c) ; (d) if is a MEC of , then ; (e) if is an ideal that lies between and an ideal , then ; and, (f) there are no containment relations among the ideals in ; is a prime divisor of }.

**[HRS1]**William Heinzer, L. J. Ratliff Jr., and Kishor Shah,*On the embedded primary components of ideals. I*, J. Algebra**167**(1994), no. 3, 724–744. MR**1287067**, 10.1006/jabr.1994.1209**[HRS2]**William Heinzer, L. J. Ratliff Jr., and Kishor Shah,*On the embedded primary components of ideals. II*, J. Pure Appl. Algebra**101**(1995), no. 2, 139–156. MR**1348032**, 10.1016/0022-4049(94)00018-E**[HRS3]**William Heinzer, L. J. Ratliff Jr., and Kishor Shah,*On the embedded primary components of ideals. III*, J. Algebra**171**(1995), no. 1, 272–293. MR**1314101**, 10.1006/jabr.1995.1012**[Mat]**Hideyuki Matsumura,*Commutative ring theory*, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR**879273****[N]**Emmy Noether,*Idealtheorie in Ringbereichen*, Math. Ann.**83**(1921), no. 1-2, 24–66 (German). MR**1511996**, 10.1007/BF01464225**[Nag]**Masayoshi Nagata,*Local rings*, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR**0155856**

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1249882-7

Article copyright:
© Copyright 1995
American Mathematical Society