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 in a commutative ring , the ring has a maximalsubring. It is also observed that if is a commutative ringwith or , then has a maximal subring. It is proved that the well-known andinteresting property of the field of the real numbers (i.e., has only one nonzero ringendomorphism) is preserved by its maximal subrings. Finally, wecharacterize submaximal ideals (an ideal of a ring iscalled submaximal if the ring 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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A. Azarang, O.A.S. Karamzadeh, Which Fields Have No Maximal Subrings?. Rend. Sem. Mat. Univ. Padova 126 (2011), pp. 213–228DOI 10.4171/RSMUP/126-12