Milnor方图中的w-凝聚性

w-Coherence in Milnor Squares

整环R称为ω-凝聚整环,是指R的每个有限型理想是有限表现型的.本文证明了ω-凝聚整环是v-凝聚整环,且若(RDTF,M)是Milnor方图,则在Ⅰ型情形,R是ω-凝聚整环当且仅当D和T都是ω-整环,且T_M是赋值环;对于Ⅱ-型情形,R是ω-凝聚整环当且仅当D是域,[F:D]<∞,M是R的有限型理想,T是ω-凝聚整环,且R_M是凝聚整环.

A domain R is called w-coherent if every finite type ideal is of finitely presented type.In this paper we show that w-coherent domains are v-coherent and if （RDTF,M） is a Milnor square,then for the case that F is the quotient field of D,R is w-coherent if and only if D and T are in-coherent and Tm is a valuation domain;and for the case that F is not the quotient field of D,R is w-coherent if and only if D is a field,[F：D]∞,M is a finite type ideal of R,T is w-coherent and Rm is coherent.