Which Fields Have No Maximal Subrings?

  • A. Azarang

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

    Chamran University, Ahvaz, Iran


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 in a commutative ring , the ring has a maximal subring. It is also observed that if is a commutative ring with or , then has a maximal subring. It is proved that the well-known and interesting property of the field of the real numbers (i.e., has only one nonzero ring endomorphism) is preserved by its maximal subrings. Finally, we characterize submaximal ideals (an ideal of a ring is called submaximal if the ring 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