Local rings whose multiplicative group is nilpotent. (Q1423769)

It is shown that the \(n\)-th term of the upper central series of the multiplicative group of a local ring \(R\) coincides with the multiplicative group of the \(n\)-th term of the upper central series of the associated Lie ring of \(R\). In particular, the multiplicative group of \(R\) is nilpotent if and only if the associated Lie ring of \(R\) is nilpotent and the nilpotency classes of both structures coincide.