In this paper we show that if (R,m) is a commutative Gorenstein local ring with maximal ideal m andM is an Artinian R-module, then depth(R) = Width(M)+ sup{i 2 N0 : Exti R(E(R/m),M) 6= 0}. Also, we prove that the following statements are equivalent: (1) R is Gorenstein. (2) R is Cohen-Macaulay and for any Artinian module M, fd(E(M)) fd(M), where E(M) is an injective envelope of M. (3) R is Cohen-Macaulay and for any finite length module M of finite injective dimension, id(F(M)) = id(M), where F(M) is a flat cover of M.
