Which Fields Have No Maximal Subrings?

Fields which have no maximal subrings are completely determined. We observe that the quotient fields of non-field domains have maximal subrings. It is shown that for each non-maximal prime ideal P in a commutative ring R , the ring R_P has a maximal subring. It is also observed that if R is a commutative ring with |\mathit{Max}(R)|>2^{\aleph_0} or |R/J(R)|>2^{2^{\aleph_0}} , then R has a maximal subring. It is proved that the well-known and interesting property of the field of the real numbers \mathbb{R} (i.e., \mathbb{R} has only one nonzero ring endomorphism) is preserved by its maximal subrings. Finally, we characterize submaximal ideals (an ideal I of a ring R is called submaximal if the ring R/I has a maximal subring) in the rings of polynomials in finitely many variables over any ring. Consequently, we give a slight generalization of Hilbert's Nullstellensatz.