LCM-splitting sets in some ring extensions

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


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