KYUNGPOOK Math. J. 40(2000), 259-263 M -cancellation Ideals P. Nasehpour and S. Yassemi Department of Mathematics, University of Tehran, P. O. Box 13145-448, Tehran, Iran. e-mail : yassemi@khayam.ut.ac.ir (2000 Mathematics Subject Classification : 13C05, 13A15, 13B25) Let R be a commutative ring with non–zero identy and let M be an R–module. An ideal a of R is called an M-cancellation ideal if whenever aP = aQ for submodules P and Q of M, then P = Q. This notion is a generalization of the notion, cancellation ideal. We use M-cancellation ideals and a generalization of Dedekind–Mertens lemma to prove that for an R–module M with ZR(M)= {0}, the following statements are equivalent : (i) Every non–zero finitely generated ideal of R is an M-cancellation ideal of R. (ii) For every f ∈ R[t] and g ∈ M[t], c(fg) = c(f )c(g). 1. Introduction Let R be a commutative ring with identity. An ideal a of R is called a can- cellation ideal if whenever ab = ac for ideals b and c of R, then b = c. A good introduction to cancellation ideals may be found in Gilmer [[3] ; section 6]. An R–module M is called cancellation module, if whenever aM = bM for ideals a and b of R, then a = b, cf. [6]. The ideal a of R is called an M -cancellation ideal if whenever aP = aQ for submodules P and Q of M , then P = Q. In the first section we characterize M -cancellation ideals. Let M be a cancellation module and let a be an ideal of R. Then a is an M -cancellation ideal of R if and only if a is locally an M –regular principal ideal of R. This result is a generalization of Anderson and Roitman in [2]. In section 2 we use M -cancellation ideals and a generalization of Dedekind– Mertens lemma to prove some results of content formulas for polynomial modules such as, for an R–module M with Z R (M )= {0}, the following statements are equivalent: (i) Every non–zero finitely generated ideal of R is an M -cancellation ideal of R. (ii) For every f ∈ R[t] and g ∈ M [t], c(fg) = c(f )c(g). (Received : October 19, 1999. Revised : 13 June, 2000) Key words and phrases : cancellation ideal, Dedekind–Mertens lemma. 259