# LCM-splitting sets in some ring extensions

@inproceedings{Dumitrescu2002LCMsplittingSI, title={LCM-splitting sets in some ring extensions}, author={T. Dumitrescu and Muhammad Zafrullah}, year={2002} }

- Published 2002
DOI:10.1090/S0002-9939-01-06301-8

Let S be a saturated multiplicative set of an integral domain D. Call S an lcm splitting set if dDS ∩ D and dD ∩ sD are principal ideals for every d ∈ D and s ∈ S. We show that if R is an R2-stable overring of D (that is, if whenever a, b ∈ D and aD ∩ bD is principal, it follows that (aD∩bD)R = aR∩bR) and if S is an lcm splitting set of D, then the saturation of S in R is an lcm splitting set in R. Consequently, if D is Noetherian and p ∈ D is a (nonzero) prime element, then p is also a prime… CONTINUE READING