On the existence of maximal subrings in commutative noetherian rings

In this note, we generalize the results of [Submaximal integral domains, Taiwanese J. Math. 17(4) (2013) 1395-1412; Which fields have no maximal subrings? Rend. Sem. Mat. Univ. Padova 126 (2011) 213-228; On the existence of maximal subrings in commutative artinian rings, J. Algebra Appl. 9(5) (2010) 771-778; On maximal subrings of commutative rings, Algebra Colloq. 19(Spec 1) (2012) 1125-1138] for the existence of maximal subrings in a commutative noetherian ring. First, we show that for determining when an infinite noetherian ring R has a maximal subring, it suffices to assume that R is an integral domain with vertical bar R/I vertical bar < vertical bar R vertical bar for each nonzero ideal I of R. We determine when the latter integral domains have maximal subrings. In particular, we show that every uncountable noetherian ring has a maximal subring.