Abstract
Let R be a commutative ring with identity. We study the concept of pointwise maximal subrings of a ring. A ring R is called a pointwise maximal subring of a ring T if \(R\subset T\) and for each \(t\in T{\setminus } R\), the ring extension \(R[t]\subseteq T\) has no proper intermediate ring. A characterization of local, integrally closed pointwise maximal subrings of a ring is given. Let G be a subgroup of the group of automorphisms of T. Then the integrally closed pointwise maximality is a G-invariant property of ring extension under some conditions. We also discuss the number of overrings and the Krull dimension of pointwise maximal subrings of a ring. The pointwise maximal subrings of the polynomial ring R[X] are also discussed.
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References
Ayache, A.: Minimal overrings of an integrally closed domain. Commun. Algebra 31(12), 5693–5714 (2003)
Azarang, A.: On maximal subrings. Far East J. Math. Sci. 32(1), 107–118 (2009)
Azarang, A., Karamzadeh, O.A.S., Namazi, A.: Hereditary properties between a ring and its maximal subrings. Ukrain. Math. J. 65(7), 981–994 (2013)
Badawi, A.: On 2-absorbing ideals of commutative rings. Bull. Austral. Math. Soc. 75(3), 417–429 (2007)
Bastida, E., Gilmer, R.: Overrings and divisorial ideals of rings of the form \(D+M\). Mich. Math. J. 20, 79–95 (1973)
Cahen, P., Picavet, G., Picavet-L’Hermitte, M.: Pointwise minimal extensions. Arab. J. Math. 7, 249–271 (2018)
Davis, D.E.: Overrings of commutative rings III: Normal pairs. Trans. Am. Math. Soc. 182, 175–185 (1973)
Dobbs, D.E.: Every commutative ring has a minimal ring extension. Commun. Algebra 34(10), 3875–3881 (2006)
Dechéne, L.I.: Adjacent extensions of rings.PhD dissertation, University of California, Riverside (1978)
Dobbs, D.E., Picavet, G., Picavet-L’Hermitte, M.: Characterizing the ring extensions that satisfy FIP or FCP. J. Algebra 371, 391–429 (2012)
Dobbs, D.E., Shapiro, J.: Descent of divisibility properties of integral domains to fixed rings. Houston J. Math. 32(2), 337–353 (2006)
Dobbs, D.E., Shapiro, J.: Normal pairs with zero-divisors. J. Algebra Appl. 10(2), 335–356 (2011)
Ferrand, D., Olivier, J.-P.: Homomorphismes minimaux d’anneaux. J. Algebra 16, 461–471 (1970)
Gilbert, M.S.: Extensions of commutative rings with linearly ordered intermediate rings. PhD dissertation, University of Tennessee, Knoxville (1996)
Gilmer Jr., R.W.: Overrings of Prüfer domains. J. Algebra 4, 331–340 (1966)
Gilmer, R., Hoffmann, J.: A characterization of Prüfer domains in terms of polynomials. Pac. J. Math. 60(1), 81–85 (1975)
Kaplansky, I.: Commutative Rings. University Chigaco Press, Chicago (1974)
Modica, M. L.: Maximal subrings. PhD Dissertation, University of Chicago (1975)
Nagata, M.: Local Rings. Interscience, New York (1962)
Papick, I.J.: Topologically defined classes of going-down domains. Trans. Am. Math. Soc. 219, 1–37 (1976a)
Papick, I.J.: Local minimal overrings. Can. J. Math. 28(4), 788–792 (1976b)
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R. Kumar: The author was supported by the SRF Grant from UGC India, Sr. No. 2061440976.
A. Gaur: The author was supported by the MATRICS Grant from DST-SERB, No. MTR/2018/000707.
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Kumar, R., Gaur, A. Pointwise maximal subrings. Beitr Algebra Geom 62, 843–856 (2021). https://doi.org/10.1007/s13366-020-00537-0
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DOI: https://doi.org/10.1007/s13366-020-00537-0