some results on the artinian cofinite modules

let i be an ideal of a commutative noetherian ring r and m be a non-zero artinian r-module with support contained in v (i). in this paper it is shown that m is i-cofinite if and only if rad(ibrj +annbrj m) = jbr j , where j := \m2suppm m and brjdenotes the j-adic comletion of r.