# ALMOST SPLITTING SETS S OF AN INTEGRAL DOMAIN D SUCH THAT DS IS A PID

• Accepted : 2011.06.19
• Published : 2011.06.30

#### Abstract

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.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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