Abstract
Let R be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring R of an integral domain S is called a maximal non valuation domain in S if R is not a valuation subring of S, and for any ring T such that R ⊂ T ⊂ S, T is a valuation subring of S. For a local domain S, the equivalence of an integrally closed maximal non VD in S and a maximal non local subring of S is established. The relation between dim(R, S) and the number of rings between R and S is given when R is a maximal non VD in S and dim(R, S) is finite. For a maximal non VD R in S such that R ⊂ R'S ⊂ S and dim(R, S) is finite, the equality of dim(R, S) and dim(R'S, S) is established.
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The authors wish to thank the referee for his valuable suggestions, comments, and corrections.
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The first author was supported by a grant from UGC India, Sr.No. 2061440976 and the second author was supported by the MATRICS grant from DST-SERB, No. MTR/2018/000707.
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Kumar, R., Gaur, A. Maximal non Valuation Domains in an Integral Domain. Czech Math J 70, 1019–1032 (2020). https://doi.org/10.21136/CMJ.2020.0098-19
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DOI: https://doi.org/10.21136/CMJ.2020.0098-19