THE STRUCTURE OF PRIME RINGS WITH MAXIMUM CONDITIONS

Abstract

course that (i,,(c))* is an integral domain. Finally, since (eS (c))* is a finite )dule over the complete local domain(,,()* and has no zero-divisors, it must a complete local domain.4 This completes the proof of our theorem. We shall conclude this note by a few remarks on the hypothesis of our theorem. observe that in our formulation of the local Theorem of Bertini, we have ;roduced the subring e and a system of regular parameters x,, ..., xt in , lile in the final assertion of the theorem only the three elements xi, x2, X3 enter ;o the picture. The question naturally arises whether it is necessary at all to ;roduce the regular local domain ( or even the system of parameters xl, ..., in t. As to the latter, one finds a measure of justification in the fact that, en in the equal-characteristic case where T9 contains a coefficient field k, an 9itrary set of three analytically independent elements xl, x2, X3 may generate an ideal of rank less than 3 and hence cannot be extended into a system of rameters in ?. This is a well-known situation where a local ring may contain as ;ubring a local ring of higher dimension, and one can easily obtain examples to ow that the local Theorem of Bertini cannot hold for an arbitrary set of three