JournalsrsmupVol. 126pp. 213–228

Which Fields Have No Maximal Subrings?

  • A. Azarang

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

    Chamran University, Ahvaz, Iran
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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 PP in a commutative ring RR, the ring RPR_P has a maximal subring. It is also observed that if RR is a commutative ring with Max(R)>20|Max(R)|>2^{\aleph_0} or R/J(R)>220|R/J(R)|>2^{2^{\aleph_0}}, then RR has a maximal subring. It is proved that the well-known and interesting property of the field of the real numbers R\mathbb{R} (i.e., R\mathbb{R} has only one nonzero ring endomorphism) is preserved by its maximal subrings. Finally, we characterize submaximal ideals (an ideal II of a ring RR is called submaximal if the ring R/IR/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