Let $ R $ be a finite commutative chain ring with invariants $ p, n, r, k, m. $ It is known that $ R $ is an extension over a Galois ring $ GR(p^n, r) $ by an Eisenstein polynomial of some degree $ k $. If $ p\nmid k, $ the enumeration of such rings is known. However, when $ p\mid k $, relatively little is known about the classification of these rings. The main purpose of this article is to investigate the classification of all finite commutative chain rings with given invariants $ p, n, r, k, m $ up to isomorphism when $ p\mid k. $ Based on the notion of j-diagram initiated by Ayoub, the number of isomorphism classes of finite (complete) chain rings with $ (p-1)\nmid k $ is determined. In addition, we study the case $ (p-1)\mid k, $ and show that the classification is strongly dependent on Eisenstein polynomials not only on $ p, n, r, k, m. $ In this case, we classify finite (incomplete) chain rings under some conditions concerning the Eisenstein polynomials. These results yield immediate corollaries for p-adic fields, coding theory and geometry.
Citation: Sami Alabiad, Yousef Alkhamees. On classification of finite commutative chain rings[J]. AIMS Mathematics, 2022, 7(2): 1742-1757. doi: 10.3934/math.2022100
Let $ R $ be a finite commutative chain ring with invariants $ p, n, r, k, m. $ It is known that $ R $ is an extension over a Galois ring $ GR(p^n, r) $ by an Eisenstein polynomial of some degree $ k $. If $ p\nmid k, $ the enumeration of such rings is known. However, when $ p\mid k $, relatively little is known about the classification of these rings. The main purpose of this article is to investigate the classification of all finite commutative chain rings with given invariants $ p, n, r, k, m $ up to isomorphism when $ p\mid k. $ Based on the notion of j-diagram initiated by Ayoub, the number of isomorphism classes of finite (complete) chain rings with $ (p-1)\nmid k $ is determined. In addition, we study the case $ (p-1)\mid k, $ and show that the classification is strongly dependent on Eisenstein polynomials not only on $ p, n, r, k, m. $ In this case, we classify finite (incomplete) chain rings under some conditions concerning the Eisenstein polynomials. These results yield immediate corollaries for p-adic fields, coding theory and geometry.
[1] | S. Alabiad, Y. Alkhamees, Recapturing the structure of group of units of any finite commutative chain ring, Symmetry, 13 (2021), 307. doi: 10.3390/sym13020307. doi: 10.3390/sym13020307 |
[2] | J. W. S. Cassels, Local fields, Cambridge University Press, 1986. doi: 10.1017/CBO9781139171885. |
[3] | C. W. Ayoub, On the group of units of certain rings, J. Number Theory, 4 (1972), 383–403. doi: 10.1016/0022-314X(72)90070-4 |
[4] | W. E. Clark, J. J. Liang, Enumeration of finite chain rings, J. Algebra, 27 (1973), 445–453. doi: 10.1016/0021-8693(73)90055-0. doi: 10.1016/0021-8693(73)90055-0 |
[5] | M. Greferath, Cyclic codes over finite rings, Discrete Math., 177 (1997), 273–277. doi: 10.1016/S0012-365X(97)00006-X. doi: 10.1016/S0012-365X(97)00006-X |
[6] | P. W. Haggard, J. O. Kiltenin, Binomial expansion modulo prime powers, Int. J. Math. Math. Sci., 3 (1980), 985261. doi: 10.1155/S0161171280000270. doi: 10.1155/S0161171280000270 |
[7] | X. D. Hou, Finite commutative chain rings, Finite Fields Appl., 7 (2001), 382–396. doi: 10.1006/ffta.2000.0317. doi: 10.1006/ffta.2000.0317 |
[8] | J. Neukirch, Local class field theory, Berlin: Springer, 1986. |
[9] | W. Klingenberg, Projective und affine Ebenen mit Nachbarelementen, Math. Z., 60 (1954), 384–406. doi: 10.1007/BF01187385. doi: 10.1007/BF01187385 |
[10] | S. Lang, Algebraic number theory, New York: Springer, 1994. doi: 10.1007/978-1-4612-0853-2. |
[11] | X. S. Lui, H. L. Lui, LCD codes over finite chain rings, Finite Fields Appl., 34 (2015), 1–19. doi: 10.1016/j.ffa.2015.01.004. doi: 10.1016/j.ffa.2015.01.004 |
[12] | B. R. McDonald, Finite rings with identity, New York: Marcel Dekker, 1974. |
[13] | R. Raghavendran, Finite associative rings, Compos. Math., 21 (1969), 195–229. |
[14] | M. J. Shi, S. X. Zhu, S. L. Yang, A class of optimal p-ary codes from one-weight codes over $F_{p}[u]/ < u^m>$, J. Franklin I., 350 (2013), 929–937. doi: 10.1016/j.jfranklin.2012.05.014. doi: 10.1016/j.jfranklin.2012.05.014 |