Coprime packedness and set theoretic complete intersections of ideals in polynomial rings

Author:
V. Erdogdu

Journal:
Proc. Amer. Math. Soc. **132** (2004), 3467-3471

MSC (2000):
Primary 13B25, 13B30, 13C15, 13C20; Secondary 13A15, 13A18

Published electronically:
July 14, 2004

MathSciNet review:
2084066

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Abstract | References | Similar Articles | Additional Information

Abstract: A ring is said to be coprimely packed if whenever is an ideal of and is a set of maximal ideals of with , then for some . Let be a ring and be the localization of at its set of monic polynomials. We prove that if is a Noetherian normal domain, then the ring is coprimely packed if and only if is a Dedekind domain with torsion ideal class group. Moreover, this is also equivalent to the condition that each proper prime ideal of is a set theoretic complete intersection. A similar result is also proved when is either a Noetherian arithmetical ring or a Bézout domain of dimension one.

**1.**James W. Brewer and William J. Heinzer,*𝑅 Noetherian implies 𝑅⟨𝑋⟩ is a Hilbert ring*, J. Algebra**67**(1980), no. 1, 204–209. MR**595028**, 10.1016/0021-8693(80)90317-8**2.**V. Erdoğdu,*Coprimely packed rings*, J. Number Theory**28**(1988), no. 1, 1–5. MR**925604**, 10.1016/0022-314X(88)90115-1**3.**V. Erdogdu,*The prime avoidance of ideals in Noetherian Hilbert rings*, Communications in Algebra,**22**(1994), 4989-4990.**4.**V. Erdoğdu,*Three notes on coprime packedness*, J. Pure Appl. Algebra**148**(2000), no. 2, 165–170. MR**1759390**, 10.1016/S0022-4049(00)00003-7**5.**V. Erdoğdu and S. McAdam,*Coprimely packed Noetherian polynomial rings*, Comm. Algebra**22**(1994), no. 15, 6459–6470. MR**1303015**, 10.1080/00927879408825200**6.**Sarah Glaz and Wolmer V. Vasconcelos,*The content of Gaussian polynomials*, J. Algebra**202**(1998), no. 1, 1–9. MR**1614237**, 10.1006/jabr.1997.7333**7.**Irving Kaplansky,*Commutative rings*, Revised edition, The University of Chicago Press, Chicago, Ill.-London, 1974. MR**0345945****8.**L. R. le Riche,*The ring 𝑅⟨𝑋⟩*, J. Algebra**67**(1980), no. 2, 327–341. MR**602067**, 10.1016/0021-8693(80)90164-7**9.**David E. Rush,*Generating ideals up to radical in Noetherian polynomial rings*, Comm. Algebra**25**(1997), no. 7, 2169–2191. MR**1451687**, 10.1080/00927879708825981

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Additional Information

**V. Erdogdu**

Affiliation:
Department of Mathematics, Istanbul Technical University, Maslak, 80626 Istanbul, Turkey

Email:
erdogdu@itu.edu.tr

DOI:
https://doi.org/10.1090/S0002-9939-04-07438-6

Keywords:
Coprime packedness,
polynomial rings,
class group,
set theoretic complete intersection

Received by editor(s):
July 17, 2002

Received by editor(s) in revised form:
June 25, 2003

Published electronically:
July 14, 2004

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2004
American Mathematical Society