A classification up to algebra isomorphism of the ramified minimal ring extensions of a principal ideal ring. (English) Zbl 1494.13011
Author’s abstract: For any (unital commutative) principal ideal ring \(A\), The author produced a set of \(A\)-algebra isomorphism class representatives of the ramified (unital minimal) ring extensions of \(A\). Special attention is paid to the case of a finite principal ideal ring. The first step in this process involves sharpening some of the author’s earlier results from the case where \(A\) is a finite special principal ideal ring. One consequence of this work is the completion of the classification, up to ring isomorphism, of the commutative (unital) rings having a unique proper (unital) subring.
Reviewer: Jebrel M. Habeb (Irbid)
MSC:
| 13B99 | Commutative ring extensions and related topics |
| 13B21 | Integral dependence in commutative rings; going up, going down |
| 13F10 | Principal ideal rings |
| 13G05 | Integral domains |
| 13M99 | Finite commutative rings |
Keywords:
commutative ring; ring extension; minimal ring extension; integral; ramified; special principa lideal ring; principal ideal ring; idealization; direct product; maximal ideal; decomposed; inert; algebra homomorphism.References:
| [1] | D. E. Dobbs, On the commutative rings with at most two proper subrings, Int. J. Math. Math. Sci., volume 2016 (2016), Article ID 6912360, 13 pages. DOI: 10.1155/2016/6912360. · Zbl 1441.13019 |
| [2] | D. E. Dobbs, Certain towers of ramified minimal ring extensions of commutative rings, Comm. Algebra, 46 (8) (2018), 3461-3495. DOI: 10.1080/00927872.2017.1412446. · Zbl 1402.13011 |
| [3] | D. E. Dobbs, A minimal ring extension of a large finite local prime ring is probably ramified, J. Algebra Appl., 19 (1) (2020), 2050015 (27 pages). DOI: 10.1142/S0219498820500152. · Zbl 1440.13044 |
| [4] | D. E. Dobbs, Characterizing finite fields via minimal ring extensions, Comm. Algebra, 47 (2019), 4945- 4957. DOI: 10.1080/00927872.2019.1603303. · Zbl 1427.13006 |
| [5] | D. E. Dobbs, On the nature and number of the minimal ring extensions of a finite commutative ring, Comm. Algebra, 48 (9) (2020), 3811-3833. DOI: 10.1080/00927872.2020.1748193. · Zbl 1457.13017 |
| [6] | D. E. Dobbs and N. Jarboui, Characterizing finite Boolean rings by using finite chains of subrings, Gulf J. Math., 10 (1) (2021), 69-94. · Zbl 1479.16019 |
| [7] | D. E. Dobbs, B. Mullins, G. Picavet and M. Picavet-L’Hermitte, On the FIP property for extensions of commutative rings, Comm. Algebra, 33 (2005), 3091-3119. · Zbl 1120.13009 |
| [8] | D. E. Dobbs and J. Shapiro, A classification of the minimal ring extensions of an integral domain, J. Algebra, 305 (2006), 185-193. · Zbl 1107.13010 |
| [9] | D. E. Dobbs and J. Shapiro, A classification of the minimal ring extensions of certain commutative rings, J. Algebra, 308 (2007), 800-821. · Zbl 1118.13004 |
| [10] | D. Ferrand and J.-P. Olivier, Homomorphismes minimaux d’anneaux, J. Algebra, 16 (1970), 461-471. · Zbl 0218.13011 |
| [11] | R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972. · Zbl 0248.13001 |
| [12] | J. A. Huckaba, Commutative Rings with Zero Divisors, Dekker, New York, 1988. · Zbl 0637.13001 |
| [13] | I. Kaplansky, Commutative Rings, rev. ed., Univ. Chicago Press, Chicago, 1974. · Zbl 0296.13001 |
| [14] | A. A. Klein, The finiteness of a ring with a finite maximal subring, Comm. Algebra, 21 (4) (1993), 1389-1392. · Zbl 0793.16009 |
| [15] | T. J. Laffey, A finiteness theorem for rings, Proc. Roy. Irish Acad. Sect. A, 92 (2) (1992), 285-288. · Zbl 0792.16005 |
| [16] | B. R. McDonald, Finite Rings with Identity, Dekker, New York, 1974. · Zbl 0294.16012 |
| [17] | J. Sato, T. Sugatani and K. Yoshida, On minimal overrings of a Noetherian domain, Comm. Algebra, 20 (1992), 1735-1746. · Zbl 0747.13017 |
| [18] | O. Zariski and P. Samuel, Commutative Algebra, Volume I, Van Nostrand, Princeton-Toronto-London, 1958 · Zbl 0081.26501 |
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