Let R be a commutative ring with unity. The notion of maximal non λ-subrings is introduced and studied. A ring R is called a maximal non λ-subring of a ring T if R ⊂ T is not a λ-extension, and for any ring S such that R ⊂ S ⊆ T, S ⊆ T is a λ-extension. We show that a maximal non λ-subring R of a field has at most two maximal ideals, and exactly two if R is integrally closed in the given field. A determination of when the classical D + M construction is a maximal non λ-domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non λ-subring. If R is a maximal non λ-subring of a field K, where R is integrally closed in K, then K is the quotient field of R and R is a Prüfer domain. The equivalence of a maximal non λ-domain and a maximal non valuation subring of a field is established under some conditions. We also discuss the number of overrings, chains of overrings, and the Krull dimension of maximal non λ-subrings of a field.
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The authors thank the referee for his fruitful comments which helped to improve the quality of the paper.
The first author was supported by Junior Research Fellowship from UGC, India.
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Kumar, R., Gaur, A. Maximal Non λ-Subrings. Czech Math J 70, 323–337 (2020). https://doi.org/10.21136/CMJ.2019.0298-18
- maximal non λ-subring
- integrally closed extension
- valuation domain