# Which Fields Have No Maximal Subrings?

### A. Azarang

Chamran University, Ahvaz, Iran### O.A.S. Karamzadeh

Chamran University, Ahvaz, Iran

## 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 $∣Max(R)∣>2_{ℵ_{0}}$ or $∣R/J(R)∣>2_{2_{ℵ}}$, then $R$ has a maximal subring. It is proved that the well-known and interesting property of the field of the real numbers $R$ (i.e., $R$ 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.

## Cite this article

A. Azarang, O.A.S. Karamzadeh, Which Fields Have No Maximal Subrings?. Rend. Sem. Mat. Univ. Padova 126 (2011), pp. 213–228

DOI 10.4171/RSMUP/126-12