Commutative rings whose factor domains are infinite (Q2796380)

The author prove that the following are equivalent for a commutative ring \(R\): {\parindent=6mm \begin{itemize} \item[1.] For each prime ideal \(P\) of \(R\), the factor ring \(R/P\) is infinite. \item [2.] For each maximal ideal \(M\) of \(R\), the factor ring \(R/M\) is infinite. \item [3.] For each minimal prime ideal \(Q\) of \(R\), the factor ring \(R/Q\) is infinite.NEWLINENEWLINEIn particular, if \(R\) is an integral domain with prime subring \(A\), then the above equivalent conditions are equivalent to the following condition too: \item [4.] Whenever \(M\) is a maximal ideal of \(R\) such that \(R/M\) is algebraic over \(A/(M\cap A)\), then \(R/M\) is not a finitely generated module over \(A/(M\cap A)\).NEWLINENEWLINE\end{itemize}}