A characterization of separable polynomials over a ring
Let R be a ring, not necessarily commutative, having a unity. Write \(S=R[X]\) for the polynomial ring in an indeterminate X satisfying \(rX=Xr\) for each \(r\in R\). Suppose that f(X)\(\in S\) is monic and that \(I=(f(X))\) is the ideal of S generated by f(X). Then f(X) is called a separable polynomial over R if \(\bar S=S/I\) is a separable ring extension over R having a free basis \(\{1,x,...,x^{n-1}\}\), where \(x=X+I\). Given \(s\in \bar S\), we define the trace t(s) to be \(\sum_{i}\pi_ i(sx^ i)\), where \(\pi_ i\) is the \(i^{th}\) projection map, and we let T be the \(n\times n\) matrix with \(t(x^ ix^ j)\) as the entry in position \((i+1,j+1)\), for \(i,j=0,1,...,n-1.\) \textit{F. DeMeyer} and \textit{E. Ingraham} [Separable algebras over commutative rings (Lect. Notes Math. 181, 1971; Zbl 0215.366)] have shown that if R is commutative then f(X) is separable over R if and only if the determinant of T is a unit in R. This theorem is generalized in the present paper. It is shown that (i) when R is finitely generated and projective over its centre, f(X) is separable if and only if T is invertible and (ii) when R is non-commutative, finitely generated and projective over its centre C, then f(X) is separable if and only if the determinant of T is a unit in C.
scientific article; zbMATH DE number 3979274