On the length of modules over Artinian local rings (Q1071064)

For an artinian module M over a local artinian commutative ring R, the author defines t(M) to be one, if \(M=0\) and to be \(\ell (R/0:M)/\ell (M)\) otherwise; \(T(M)=\sup \{t(N)| N\subseteq M\}\) and \(r(M)=\ell (socle(M)),\) where \(\ell =length\). He first shows that if \(M\neq 0\), then \(1/r(M)\leq t(M)\leq r(M),\) whence \(1\leq T(M)\leq r(M).\) He then shows that \(T(M)=r(M)\) if and only if \(r(M)=1\). An example shows that T(R) can be made close to any positive integer, by judicious choice of R. Some scattered results are given on rings R with \(T(R)=1\) and ideals I with t(I)\(\leq 1\). Some interesting problems are stated at the end.