LCM-splitting sets in some ring extensions

Authors:
Tiberiu Dumitrescu and Muhammad Zafrullah

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1639-1644

MSC (2000):
Primary 13A05, 13A15; Secondary 13B02, 13B22

DOI:
https://doi.org/10.1090/S0002-9939-01-06301-8

Published electronically:
November 15, 2001

MathSciNet review:
1887010

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a saturated multiplicative set of an integral domain . Call an lcm splitting set if and are principal ideals for every and . We show that if is an -stable overring of (that is, if whenever and is principal, it follows that and if is an lcm splitting set of , then the saturation of in is an lcm splitting set in . Consequently, if is Noetherian and is a (nonzero) prime element, then is also a prime element of the integral closure of . Also, if is Noetherian, is generated by prime elements of and if the integral closure of is a UFD, then so is the integral closure of .

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Additional Information

**Tiberiu Dumitrescu**

Affiliation:
Facultatea de Matematică, Universitatea Bucureşti, Str. Academiei 14, Bucharest, RO-70190, Romania

Email:
tiberiu@al.math.unibuc.ro

**Muhammad Zafrullah**

Affiliation:
Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085

Email:
zufrmuha@isu.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-06301-8

Keywords:
lcm-splitting set,
$R_{2}$-stable overring,
Noetherian domain

Received by editor(s):
May 24, 2000

Received by editor(s) in revised form:
January 15, 2001

Published electronically:
November 15, 2001

Additional Notes:
The authors gratefully acknowledge the referee’s interest in improving the presentation of this paper.

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 2001
American Mathematical Society