Gröbner Bases and Normal Forms in a Subring of the Power Series Ring on Countably Many Variables

If K is a field, let the ring R ′ consist of finite sums of homogeneous elements in R = K[[x1,x2,x3,...]]. Then, R ′ contains M, the free semi-group on the countable set of variables {x1,x2,x3,...}. In this paper, we generalize the notion of admissible order from finitely generated sub-monoids of M to M itself; assume that> is such an admissible order on M. We show that we can define leading power products, with respect to>, of elements in R ′ , and thus the initial ideal gr(I) of an arbitrary ideal I ⊂ R ′.IfI is what we call a locally finitely generated ideal, then we show that gr(I) is also locally finitely generated; this implies that I has a finite truncated Gröbner basis up to any total degree. We give an example of a finitely generated homogeneous ideal which has a non-finitely generated initial ideal with respect to the lexicographic initial order>lex on M. c ○ 1998 Academic Press Limited 1