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

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

MathSciNet review:
1249882

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]**W. Heinzer, L. J. Ratliff, Jr., and K. Shah,*On the embedded primary components of ideals*(I), J. Algebra**167**(1994), 724-744. MR**1287067 (95f:13023)****[HRS2]**-,*On the embedded primary components of ideals*(II), J. Pure Appl. Algebra (to appear). MR**1348032 (96h:13046)****[HRS3]**-,*On the embedded primary components of ideals*(III), J. Algebra (to appear). MR**1314101 (96h:13047)****[Mat]**H. Matsumura,*Commutative ring theory*, Cambridge Studies in Advanced Mathematics, No. 8, Cambridge Univ. Press, Cambridge, 1986. MR**879273 (88h:13001)****[N]**E. Noether,*Idealtheorie in Ringbereichen*, Math. Ann.**83**(1921), 24-66. MR**1511996****[Nag]**M. Nagata,*Local rings*, Interscience Tracts No. 13, Interscience, New York, 1961. MR**0155856 (27:5790)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
13E05,
13H99

Retrieve articles in all journals with MSC: 13E05, 13H99

Additional Information

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

Article copyright:
© Copyright 1995
American Mathematical Society