a b s t r a c tIn this work, the cardinality of the minimal R-covers of finite rings with respect to the RT-metric is established. By generalizing the result in Nakaoka and dos Santos (2010) [1], the minimal cardinalities of 0-short coverings of finite chain rings are calculated. The connection between R-short coverings of rings with respect to the RT-metric and the 0-short coverings of rings is demonstrated, and with the help of this connection, the problem of finding the minimal cardinalities of R-short coverings of finite chain rings is solved.

In this study, a representation of the group ring ZS3 is obtained by using a faithful irreducible representation of S3 of degree 2. In ZS3, all torsion units are expressed in terms of parameters by means of this representation. It is shown that any torsion unit in U1 (ZS3) can be expressed in terms of two parameters.

Abstract. The structure of V (Z(ZAn)) is known when n ≤ 6. If n = 5 or 6, then a complete set of generators of V (Z(ZAn)) has been determined. In this study, it was shown that V (Z(ZAn)) is trivial when n = 7, 8 or 9 and it is generated by a single unit u when n = 10 or 11. This unit u is characterized explicitly for n = 10 or 11 by using irreducible characters of An.

Abstract. Describing the group of units U (ZG) of the integral group ring ZG, for a finite group G, is a classical and open problem. In this note, we show that, where T = a, b : a 6 = 1, a 3 = b 2 , ba = a 5 b and F 97 , F 5 are free groups of ranks 97 and 5, respectively.