Maximal non -subrings
Let be a commutative ring with unity. The notion of maximal non -subrings is introduced and studied. A ring is called a maximal non -subring of a ring if is not a -extension, and for any ring such that , is a -extension. We show that a maximal non -subring of a field has at most two maximal ideals, and exactly two if is integrally closed in the given field. A determination of when the classical construction is a maximal non -domain is given. A necessary condition is given...