Tensor rings under field extensions
Let \(K\) be a skew field and \(k\) a central subfield. The author considers the tensor ring \(R=K_k\langle X\rangle\) on a set \(X\), defined as the \(K\)-ring generated by \(X\) with defining relations \(\alpha x=x\alpha\) (\(\alpha\in k\), \(x\in X\)). This ring \(R\) is always a fir and the author investigates the ring \(R_F=R\otimes_kF\) obtained by extending the base field \(k\) to \(F\). If \(U\) denotes the universal skew field of fractions of \(R\), then as is well known, \(U_F\) for a finite extension \(F/k\) is a full matrix ring over a skew field. The author proves that when \(F/k\) is a Galois extension of finite degree \(n\), then there is an \(n\)-element subset \(\Sigma\) of \(R\) such that the localization \(S\) of \(R\) at \(\Sigma\) has the property that \(S_F\) is an \(n\times n\) matrix ring over a fir. -- Secondly, let \(k\) have prime characteristic \(p\) and let \(F\) be a purely inseparable extension of degree \(p\). Then \(S_F\), where \(S\) is formed by adjoining the inverse of a single element to \(R\), is a \(p\times p\) matrix ring over a fir. For the proof he constructs in each case the matrix units, taking first the case when \(X\) has a single element. In the final ring he compares the base rings of \(U_F\) and \(S_F\), showing that the latter is a fir with the former as its universal skew field of fractions.
scientific article; zbMATH DE number 1525276