SIMPLE COMPLETE IDEALS IN 2-DIMENSIONAL REGULAR LOCAL-RINGS

In a two-dimensional regular local ring (R, m), it is known that there exists a unique complete ideal I adjacent to a given simple complete m-primary ideal J from above. In this paper it is shown that there are infinitely many simple complete m-primary ideals adjacent to a given simple complete m-primary ideal J (not equal m) from below whose order s are the same as that of J and that there exists a unique complete m- primary ideal adjacent to J from below whose order is one bigger than that of J. We also show that these are all the complete ideals adjacen t to J from below. It is known that there is a unique prime divisor w and a unique infinitely near point S of R associated to a given simple complete m-primary ideal J. As a corollary of the main theorem, we ob tain one-to-one correspondences between the set of simple m-primary co mplete ideals adjacent to J from below, the set of first neighborhood prime divisors of w, and the set of first quadratic transformations of S.