Let ; let and . Without loss of generality, and . Let N=max{m,n}.
Since is an ideal in R, we see that , and . Thus I is an ideal in R.
Assume every ideal in R is finitely generated and consider I in (1). It follows that I is finitely generated by . Each for i=1,2,...,m lies in one of the chains of I, say . Let . Then for all i. This implies . Thus, and for all .2.) Prove that if every ideals of R is finitely generated, then exist a number such that