Abstract
Let R be a commutative ring with unity. The notion of maximal non chained subrings of a ring and maximal non \(\phi \)-chained subrings of a ring is introduced which generalizes the concept of maximal non valuation subrings of a domain. A ring R is said to be a maximal non chained (resp., \(\phi \)-chained) subring of S if R is a proper subring of S, R is not a chained (resp., \(\phi \)-chained) ring and every subring of S which contains R properly is a chained (resp., \(\phi \)-chained) ring. We study the properties and characterizations of a maximal non chained (\(\phi \)-chained) subring of a ring. Examples of a maximal non \(\phi \)-chained subring which is not a maximal non chained subring and a maximal non chained subring which is not a maximal non \(\phi \)-chained subring are also given to strengthen the concept.
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A. Gaur: The author was supported by the MATRICS Grant from DST-SERB India, No. MTR/2018/000707. R. Kumar: The author was supported by a Grant from UGC India, Sr. No. 22/06/2014(i)EU-V.
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Gaur, A., Kumar, R. Maximal Non \(\phi \)-Chained Rings and Maximal Non Chained Rings. Results Math 74, 121 (2019). https://doi.org/10.1007/s00025-019-1043-6
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DOI: https://doi.org/10.1007/s00025-019-1043-6