Finite rings and Wilson's theorem

Finite rings and Wilson's theorem

In this paper we consider the product of all elements in the group of units in a finite ring and we generalize Wilson s theorem to finite rings. As an application, we study some generalizations of Wilson s theorem on residually finite Dedekind domains. And we also give some examples for such rings. Moreover we study some generalizations of Wilson s theorem on rings of matrices over a finite commutative ring.

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  • [1] Andr´as, S.: A combinatorial generalization of Wilson’s theorem, Australas. J. Combin. 49, 265-272 (2011).
  • [2] Dicson, L.E.: History of the Theory of Numbers, Volume 1, Chelsea Publishing Company, New York, 1952.
  • [3] Gilmer, R.W.Jr.: Finite rings having a cyclic multiplicative group of units, Amer. J. Math. 85, 447-452 (1963).
  • [4] Ireland, K. and Rosen, M.: A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, New York, 1981.
  • [5] Laˇsˇs´ak, M.: Wilson’s theorem in algebraic number fields, Math. Slovaca 50, no. 3, 303-314 (2000).
  • [6] McDonald, B.R.: Finite Rings With Identity, Pure and Applied Mathematics, Vol. 28, Marcel Dekker, Inc., New York, 1974.
  • [7] Rotman, J.J.: An Introduction to the Theory of Groups, Fourth edition, Graduate Texts in Mathematics, Vol. 148, Springer-Verlag, New York, 1995.
  • [8] Tripathi, A.: A combinatorial proof of Wilson’s theorem, Ars Combin. 80, 201-204 (2006).