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European Mathematical Society Publishing House
2016-09-19 17:05:48
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2013)
126
2011
0
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 $P$ 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 $|Max(R)|>2^{\aleph_0}$ or $|R/J(R)|>2^{2^{\aleph_0}}$, then $R$ has a maximal subring. It is proved that the well-known and interesting property of the field of the real numbers $\mathbb{R}$ (i.e., $\mathbb{R}$ 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.
General
213
228
10.4171/RSMUP/126-12
http://www.ems-ph.org/doi/10.4171/RSMUP/126-12