Choosing generators of modules over local rings (Q1340668)

Let \(R\) be a local Noetherian ring and let \({\mathfrak m}\) be the maximal ideal in \(R\). \(x \in {\mathfrak m}\) has the degree \(k\) if \(x\) is an element of \({\mathfrak m}^ k\)-th power but not in the \({\mathfrak m}^{k+1}\)-th power. In Manuscr. Math. 62, No. 3, 337-340 (1988; Zbl 0666.13011), the author proved that if \(x_ 1, x_ 2, \dots x_ n\) and \(y_ 1, y_ 2, \dots, y_ n\) are two minimal bases of an ideal \(I\) both having maximal degree- sum, then if \(k\) of the \(x_ i\) are of degree \(r\) then also \(k\) of the \(y_ i\) are of degree \(r\). In this note, the author extends these results to finitely generated modules over local rings.