Analogue of G. Higman's theorem for commutative torsion-free rings (Q1271979)

Let \(R\oplus_Z\mathbb{Q}\) be a finite dimensional rational algebra of a torsionfree ring \(R\), and let \(T\oplus N\) be Wedderburn's semidirect decomposition of \(R\oplus_Z\mathbb{Q}\). If \(R\) is commutative and if \(R\) has an almost regular automorphism of prime order \(p\), then \(N\) is nilpotent of index \(p\) and \(T_R\) is an algebra of one of the following types: (1) a direct sum of \(p\) copies of \(\mathbb{Q}\), (2) a normal extension of degree \(p\) of \(\mathbb{Q}\), or (3) \(\mathbb{Q}\). If \(R\oplus_Z\mathbb{Q}\) is a skew-field with a nontrivial automorphism, then \(R\) has an almost regular automorphism of prime order \(p\) if and only if \(R\oplus_Z\mathbb{Q}\) is a normal extension of prime degree \(p\) over the field of rationals \(\mathbb{Q}\).