Let \(G\) be a groupoid with a partially defined binary operation as given by D. Bagio and the authors [J. Algebra Appl. 9, No. 3, 501-517 (2010; Zbl 1227.16026)]. For each \(g\in G\), denote \(g^{-1}g\) by \(d(g)\) and \(gg^{-1}\) by \(r(g)\) for a unique \(g^{-1}\in G\). Let \(R\) be an algebra over a commutative ring \(K\). An action of \(G\) on \(R\) is a pair \(\beta=(\{E_g\}_{g\in G},\{\beta_g\}_{g\in G})\) such that \(E_g=E_{r(g)}\) is an ideal of \(R\), \(\beta_g\colon E_{g^{-1}}\to E_g\) is an isomorphism of \(K\)-algebras such that (1) \(\beta_e\) is the identity map of \(E_e\) for all \(e\in G_0\) (\(=\{e=d(g)=r(g^{-1})\}\)) for some \(g\in G\) and (2) \(\beta_g\beta_h(r)=\beta_{gh}(r)\) for all \((g,h)\in G^2\) and \(r\in E_{h^{-1}}=E_{(gh)^{-1}}\) where \(G^2=\{(g,h)\in G\times G\mid gh\in G\}\). The skew groupoid ring \(A=R*_\beta G=\bigoplus_{g\in G}E_g\delta_g\) such that \((x\delta_g)(y\delta_h)=x\beta_g(y)\delta_{gh}\) if \((g,h)\in G^2\) or \(0\) otherwise for all \((g,h)\in G^2\), \(x\in E_g\), \(y\in E_h\). If \(G\) is finite, \(R=\bigoplus_{e\in G_0}E_e\) with identity \(1_g\) and \(R^\beta=\{r\in R\mid\beta_g(r1_{g^{-1}})=r1_g\) for all \(g\in G\}\). Then \(R\) is called a \(\beta\)-Galois extension if there exist \(\{x_i,y_i\in R\), \(i=1,\ldots,m\) for some integer \(m\}\) such that \(\sum_ix_i\beta_g(y_i1_{g^{-1}})=\delta_{e,g}1_e\) for all \(e\in G_0\) and \(g\in G\).
The authors show some equivalent conditions for a \(\beta\)-Galois extension \(R\) of \(R^\beta\) which is Azumaya over \(C(A)\), the center of \(A\), in terms of the skew groupoid ring \(A\). Moreover, assume that \(A\) is Azumaya with \(C(A)\subset R\). An intrinsic description of the structure of \(A\) is given. Let \(R'\) be the centralizer of \(R^\beta\) in \(R\) such that \(R^{\prime\beta}1_e=(E'_e)^{G_e}\) where \(E'_e=R'1_e\) for all \(e\in G_0\). Then \(R'*_\beta G\cong\text{End}(R')_{R^{\prime\beta}}\) and \(A\cong R^\beta\otimes_{C(A)}(R'*_\beta G)\) as \(C(A)\)-algebras, and \(R'=\bigoplus_{e\in G_0}E'_e\) such that \(E'_e\) is a direct sum of suitable central Galois algebras and commutative Galois extensions.