On weak center Galois extensions of rings (Q5936429)

The purpose of the paper is to give a characterization and the structure of a weak center Galois extension \(B\), a ring with \(1\), with group \(G=\{g_1=1,g_2,\dots,g_n\}\) a finite automorphism group of \(B\), of order \(n\).NEWLINENEWLINENEWLINEIt is shown that \(B\) is a weak center Galois extension with group \(G\) if and only if for each \(g_i\neq 1\) in \(G\) there exists an idempotent \(e_i\) in \(C\) and \(\{b_ke_i\in Be_i\); \(c_ke_i\in Ce_i\), \(k=1,2,\dots,m\}\) such that \(\sum_{k=1}^m b_ke_ig_i(c_ke_i)=\delta_{1,g_i}e_i\) and \(g_i\) restricted to \(C(1-e_i)\) is an identity. The structure of a weak center Galois extension with group \(G\) is also given -- it is the sum of \(T_i\)-Galois extensions for some \(T_i\subset G\). Finally, they construct an example of a weak center Galois extension with group \(G\), which is not a Galois extension.