LINEAR GROWTH OF PRIMARY DECOMPOSITIONS OF MODULES AND INTEGRAL CLOSURES

Abstract

Let be a commutative Noetherian ring with identity. I.Swanson proved in[10] that every ideal in every commutative Noetherian ring has linear growth of primary decompositions. Later, in[8], R.Y. Sharp generalized this result to finitely generated modules over . In the first section we present another (may be simpler) proof of this generalized result. Sharp also proved in[7] that every proper ideal in has linear growth of primary decompositions for integral closures of ideals. We extend, in the last section, this result to finitely generated modules over .