Introduction: Throughout this paper, \(B\) will mean a ring with identity element 1, \(Z\) the center of \(B\), \(G\) a finite group of automorphisms of \(B\), \(B^G\) the set of all elements in \(B\) fixed under \(G\). A ring extension \(T/S\) is called a separable extension, if the \(T\)-\(T\)-homomorphism of \(T\otimes_ST\) onto \(T\) defined by \(a\otimes b\to ab\) splits, and \(T/S\) is called an \(H\)-separable extension, if \(T\otimes_ST\) is \(T\)-\(T\)-isomorphic to a direct summand of a finite direct sum of copies of \(T\). As is well known every \(H\)-separable extension is a separable extension.
Let \(\Delta=\Delta(B,G,f)\) be a crossed product with a free basis \(\{u_\sigma\mid\sigma\in G\) and \(u_1=1\}\) over \(B\) and the multiplication is given by \(u_\sigma b=\sigma(b)u_\sigma\) and \(u_\sigma u_\tau=f(\sigma,\tau)u_{\sigma\tau}\) for \(b\in B\) and \(\sigma,\tau\in G\), where \(f\) is a factor set from \(G\times G\) to \(U(Z^G)\) such that \(f(\sigma,\tau)f(\sigma\tau,\rho)=f(\tau,\rho)f(\sigma,\tau\rho)\).
We present several theorems which assert that a separable extension with some condition is an \(H\)-separable extension.