Characterizations of taut semi-local rings

Summary It is proved that the following statements are equivalent for semi-local domain R:1) R is taut (i.e., for each non-maximal prime ideal P in R, height P+depth P=altitude R).2) Every integral domain which contains and is integral over R is taut.3) R[1/b]. satisfies the second chain condition for prime ideals (s.c.c.), for each non-zero b in the Jacobson radical J of R.4) R[1/b] satisfies the first chain condition for prime ideals (f.c.c.), for some non-zero b in J.5) For each depth one prime ideal P in R, R_P satisfies the s.c.c. and height P=altitude R−1.6) R(X) is taut, where X is an indeterminate.7) For each pair of analytically independent elements b, c in R, R(c/b) is taut and altitude R(c/b)=altitude R−1.8) Each maximal set of analytically independent elements in R contains either one element or altitude R elements. Much of the theorem is then generalized (with suitable modifications) to rings which contain and are integral over a taut semi-local ring. Entrata in Redazione il 5 dicembre 1975. Research on this paper was supported in part by the National Science Foundation grant NSF GP-28939-1.