Locally nilpotent derivations on modules (Q839304)

Let \(B\) be a ring and \(\partial\) be a locally nilpotent derivation on \(B\), and let \(A:=\ker(\partial)\). Then \((M,\partial_M)\) is defined as a pre-\(\partial\)-module if \(M\) is a \(B\)-module and \(\partial_M\) satisfies \(\partial_M(bm)=\partial(b)m+b\partial_M(m)\) for each \(b\in B, m\in M\). If \(\delta_M\) is locally nilpotent, then the prefix ``pre'' is removed. The module \(M_0:=\ker(\partial_M)\) of a \(\delta\)-module \(M\) is studied, in particular its (non)finite generation as a (1) \(B\)- or (2) \(A\)-module. Amongst others, the author gives affirmative answers to (1) in case \(M\) is a free \(B\)-module of rank one or if \(\partial\) has a slice. A negative answer to (1) is given for the case that \(M\) is a projective \(B\)-module of rank one. The problem (2) is affirmatively answered if \(M\) is a free \(B\)-module of rank one, or if \(\partial\) has a slice.