## Abstract

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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## References

- [1]
*A. Ayache, A. Jaballah*: Residually algebraic pairs of rings. Math. Z.*225*(1997), 49–65. - [2]
*A. Azarang*: On maximal subrings. Far East J. Math. Sci.*32*(2009), 107–118. - [3]
*A. Azarang, O. A. S. Karamzadeh*: On the existence of maximal subrings in commutative Artinian rings. J. Algebra Appl.*9*(2010), 771–778. - [4]
*A. Azarang, G. Oman*: Commutative rings with infinitely many maximal subrings. J. Algebra Appl.*13*(2014), Article ID 1450037, 29 pages. - [5]
*A. Badawi*: On 2-absorbing ideals of commutative rings. Bull. Aust. Math. Soc.*75*(2007), 417–429. - [6]
*E. Bastida, R. Gilmer*: Overrings and divisorial ideals of rings of the form*D*+*M*. Mich. Math. J.*20*(1973), 79–95. - [7]
*M. Ben Nasr, N. Jarboui*: On maximal non-valuation subrings. Houston J. Math.*37*(2011), 47–59. - [8]
*D. E. Davis*: Overrings of commutative rings III: Normal pairs. Trans. Am. Math. Soc.*182*(1973), 175–185. - [9]
*D. E. Dobbs*: On INC-extensions and polynomials with unit content. Can. Math. Bull.*23*(1980), 37–42. - [10]
*D. E. Dobbs, G. Picavet, M. Picavet-L’Hermitte*: Characterizing the ring extensions that satisfy FIP or FCP. J. Algebra*371*(2012), 391–429. - [11]
*D. E. Dobbs, J. Shapiro*: Normal pairs with zero-divisors. J. Algebra Appl.*10*(2011), 335–356. - [12]
*E. G. Evans, Jr.*: A generalization of Zariski’s main theorem. Proc. Am. Math. Soc.*26*(1970), 45–48. - [13]
*M. S. Gilbert*: Extensions of Commutative Rings with Linearly Ordered Intermediate Rings. PhD Thesis. University of Tennessee, Knoxville, 1996. Available at https://search.proquest.com/docview/304271872?accountid=8179. - [14]
*R. W. Gilmer, Jr.*: Overrings of Prüfer domains. J. Algebra*4*(1966), 331–340. - [15]
*R. Gilmer*: Some finiteness conditions on the set of overrings of an integral domain. Proc. Am. Math. Soc.*131*(2003), 2337–2346. - [16]
*R. W. Gilmer, Jr., J. F. Hoffmann*: A characterization of Prüfer domains in terms of polynomials. Pac. J. Math.*60*(1975), 81–85. - [17]
*A. Jaballah*: Maximal non-Prüfer and maximal non-integrally closed subrings of a field. J. Algebra Appl.*11*(2012), Article ID 1250041, 18 pages. - [18]
*I. Kaplansky*: Commutative Rings. University of Chicago Press, Chicago, 1974. - [19]
*R. Kumar, A. Gaur*: On*λ*-extensions of commutative rings. J. Algebra Appl.*17*(2018), Article ID 1850063, 9 pages. - [20]
*I. J. Papick*: Topologically defined classes of going-down domains. Trans. Am. Math. Soc.*219*(1976), 1–37.

## Acknowledgment

The authors thank the referee for his fruitful comments which helped to improve the quality of the paper.

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## Additional information

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

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### Keywords

- maximal non λ-subring
- λ-extension
- integrally closed extension
- valuation domain

### MSC 2010

- 13B02
- 13B22
- 13A18