Some Types of Simple Ring Extensions 1
نویسنده
چکیده
Throughout the present article, we understand by a ring a commutative ring with identity, and by an over-ring of a ring R a ring containing R and having the same identity with R. Let T be an over-ring of a ring R. If T is generated by a single element over R and if •v is a homomorphism of T into a ring T', then •vT is generated by a single element over •vR. In õ 1, we discuss some results related to the converse of this fact, and we mainly concern with the case where T is integral over R. In õ2, we deal with a special type of ring extensions which has some algebro-geometric meaning in connection with birational correspondences, and our results include a criterion for such an extension to be generated by a single element. The writer wants to express his thanks to Professor H. Hijikata and Professor W. Heinzer for their discussions with the writer on the topics in the present article. He is also grateful to the referee for the critical reading of the manuscript. õ 1. Integrally dependent case. Obviously, the following holds: LEMMA 1.1 Let A be an ideal of an over-ring T of a ring R. If T is generated by a single element over R, then so is T/A over R/(A Ch R). The converse of this is obviously false and we want to discuss some cases where the converse is true. We shall begin with the case where R is a field and then we shall discuss its generalization to the case where R is a quasi-semi-local ring. THEOREM 1.2. Let R be a fieM and assume that its over-ring T is a noetherian local ring with maximal ideal M and that T is integral over R. Then: (1) Assume that TIM is a separable algebraic extension of a finite degree over R, then T is generated by a single element over R if and only if M is principal.
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تاریخ انتشار 2004