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Let be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring of an integral domain is called a maximal non valuation domain in if is not a valuation subring of , and for any ring such that , is a valuation subring of . For a local domain , the equivalence of an integrally closed maximal non VD in and a maximal non local subring of is established. The relation between and the number...
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...
The notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let be an extension of domains. Then is called a maximal non-pseudovaluation subring of if is not a pseudovaluation subring of , and for any ring such that , is a pseudovaluation subring of . We show that if is not local, then there no such exists between and . We also characterize maximal non-pseudovaluation subrings of a local integral domain.
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