Abstract
Let be a commutative Noetherian ring with identity. I.Swanson proved inCitation[10] that every ideal in every commutative Noetherian ring has linear growth of primary decompositions. Later, inCitation[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 inCitation[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
.