Korean J. Math. Vol. 19 No. 2 (2011) pp.163-169
ALMOST SPLITTING SETS S OF AN INTEGRAL DOMAIN D SUCH THAT $D_S$ IS A PID
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Let D be an integral domain, S be a multiplicative subset of D such that DS is a PID, and D[X] be the polynomial ring over D. We show that S is an almost splitting set in D if and only if every nonzero prime ideal of D disjoint from S contains a primary element. We use this result to give a simple proof of the known result that D is a UMT-domain and Cl(D[X]) is torsion if and only if each upper to zero in D[X] contains a primary element.
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