Abstract
It is shown that if R is a ring, p a prime element of an integral domain with
and
then R has a conch maximal subring (see [14]). We prove that either a ring R has a conch maximal subring or
for each subring S of R (i.e., each subring of R is closed with respect to taking inverse, see [25]). In particular, either R has a conch maximal subring or U(R) is integral over the prime subring of R. We observe that if R is an integral domain with
then either R has a maximal subring or
and in particular if in addition dim(R) = 1, then R has a maximal subring. If
is an integral ring extension,
then we prove that whenever R has a conch maximal subring S with
then T has a conch maximal subring V such that
and
It is shown that if K is an algebraically closed field which is not algebraic over its prime subring and R is affine ring over K, then for each prime ideal P of R with
there exists a maximal subring S of R with
If R is a normal affine integral domain over a field K, then we prove that R is an integrally closed maximal subring of a ring T if and only if dim(R) = 1 and in particular in this case
Acknowledgement
The author would like to thank the referee for helpful comments which improve the paper.