hamidreza dorbidi

A chain of proper ideals of a commutative ring R like I_0 <...< I_n such that (I_j : I_j+1) = I_j is said to be a good chain of length n. Define ght(I) as the supreme of length of good chains in I. Then gr(I) <= ght(I) and ht(P) <= ght(P) for every prime ideal P(Here gr(I) and ht(I) denote the grade and height of an ideal). Also if R is a Noetherian ring, then gr(I)<= ht(I). We say R is a good ring if gr(I) = ght(I) for every ideal I. We prove every Cohen-Macaulay ring is a good ring. Also if R is a one dimensional noetherian ring, then R is a Cohen-Macaulay ring. Also we give examples of non noetherian rings such that ht(I) < ght(I).