On generalizations of separable polynomials over rings. (Q1946863)

Based on a well-known characterization of a separable (resp. quasi-separable) extension of a ring, the authors define and investigate a generalization of this notion. For a ring extension \(S/R\), it is known that \(S/R\) is separable (resp. quasi-separable) iff for every \(S\)-bimodule \(M\), every \(R\)-derivation of \(S\) to \(M\) is inner (resp. every central \(R\)-derivation of \(S\) to \(M\) is zero). An extension \(S/R\) is then called `weakly separable' (resp. `weakly quasi-separable') if every \(R\)-derivation of \(S\) to \(S\) is inner (resp. if every central \(R\)-derivation of \(S\) to \(S\) is zero). An example is given to show that these notions are in fact more general. For a commutative ring \(R\), a monic polynomial \(f(X)\) over \(R\) is called weakly separable (resp. weakly quasi-separable) if the extension \(R[x]/R\) is weakly separable (resp. weakly quasi-separable) where \(R[x]\) is the quotient \(R[x]=R[X]/(f(X))\) with \(x=X+(f(X))\). It is shown that \(f(X)=X^n-aX-b\) is weakly separable in \(R[X]\) if and only if the discriminant of \(f(X)\) is a nonzero-divisor in \(R\). The authors then investigate the structure of these newly defined notions for the quotient ring \(R[x,*]=R[X,*]/f(X)R[X,*]\) where \(R[X,*]\) is a skew polynomial ring and \(*\) is a ring automorphism or a derivation of \(R\).