Krull dimension of polynomial and power series rings (Q2895445)

In their paper [``Krull dimension of polynomial rings'', Conf. commutat. Algebra, Lawrence, Kansas 1972, Lect. Notes Math. 311, 26--45 (1973; Zbl 0249.13004)], \textit{J. W. Brewer, P. R. Montgomery, E. A. Rutter} and \textit{W. J. Heinzer} proved the following theorem: Let \(R\) be a commutative ring with 1. Let \(Q\) be a prime ideal of the ring \(R[x_1,\dots,x_n]\) and \(P=Q\cap R\). Then \(\text{ht}Q=\text{ht}P[x_1,\dots,x_n]+\text{ht}(Q/P[x_1,\dots,x_n])\leq \text{ht} P[x_1,\dots,x_n]+n\). In the present paper, the author recalls the preceding theorem with its proof. Then he applies it to give elementary proofs of several well known results concerning Krull dimension for polynomial ring. As an example, he proves that for a commutative Noetherian or semihereditary ring \(R\) with 1, \(\dim R[x_1,\dots,x_n]=n+\dim R\). The author ends by a slight comparison with the dimension of power series ring where the situation is more complicated and very different from the polynomial case.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13006].