# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 126, 2011, pp. 213–228**

**DOI: 10.4171/RSMUP/126-12**

Published online: 2011-12-31

Which Fields Have No Maximal Subrings?

A. Azarang^{[1]}and O.A.S. Karamzadeh

^{[2]}(1) Chamran University, Ahvaz, Iran

(2) Chamran University, Ahvaz, Iran

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

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Azarang A., Karamzadeh O.A.S.: Which Fields Have No Maximal Subrings?. *Rend. Sem. Mat. Univ. Padova* 126 (2011), 213-228. doi: 10.4171/RSMUP/126-12