In the $d$ variable polynomial ring $R=k[x_{1},\cdots,x_{d}]$ show that $0, x_{1}R, (x_{1},x_{2})R, \cdots , (x_{1},x_{2},...,x_{d})R$ is a strictly increasing sequence of prime ideals and there is no longer such chain. how do i prove this claim.
Well, i am thinking that for this problem this result may be helpful: for a ring, $R$, to be Noetherian, one formulation dictates that any ascending chain of ideals in R terminates. But i just can't get going with it.