Which Fields Have No Maximal Subrings?

Abstract

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 $|Max(R)|>2^{{\aleph }_0}$ 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.