A note on the coefficient rings of polynomial rings (Q581606)

The investigation in the present paper is motivated by the following question posed by \textit{S. S. Abhyankar}, \textit{W. Heinzer} and \textit{P. Eakin} [J. Algebra 23, 310-342 (1972; Zbl 0255.13008)]. If \(A[X_ 1,...,X_ n]=B[Y_ 1,...,Y_ n]\), do there exist isomorphisms of A into B and B into A? Using Noether's normalisation lemma, the author proves the following result: Let A and B be two reduced rings with finitely many minimal prime ideals. If the polynomial rings \(A[X_ 1,...,X_ n]\) and \(B[Y_ 1,...,Y_ n]\) are equal, then there exists an injection \(\Phi:\quad A\to B\) such that B is finitely generated over \(\Phi\) (A). There are several misprints in the paper.