Partial actions and quotient rings. (Q2891211)

The author relates the notion of partial actions with Martindale rings of quotients. Let \(G\) be a group and \(R\) a unital \(k\)-algebra, where \(k\) is a commutative ring. A partial action \(\alpha\) of \(G\) on \(R\), denoted by \((R,\alpha)\), is a collection of ideals \(S_g\) of \(R\), \(g\in G\), and isomorphisms of (non-necessarily unital) \(k\)-algebras \(\alpha_g\colon S_{g^{-1}}\to S_g\) such that for all \(g,h\in G\) the following statements hold:NEWLINENEWLINE \(\bullet\) \(S_1=R\) and \(\alpha_1\) is the identity mapping of \(R\).NEWLINENEWLINE \(\bullet\) \(S_{(gh)^{-1}}\supseteq\alpha_h^{-1}(S_h\cap S_{g^{-1}})\).NEWLINENEWLINE \(\bullet\) \(\alpha_g\circ\alpha_h(x)=\alpha_{gh}(x),\) for any \(x\in\alpha_h^{-1}(S_h\cap S_{g^{-1}})\).NEWLINENEWLINE As examples of partial actions, restrictions of global actions to ideals can be considered: If a group \(G\) acts on an algebra \(T\) by automorphisms \(\beta_g\colon R\to R\) for every \(g\in G\), and \(R\) is an ideal of \(T\), then \(S_g:=R\cap\beta_g(R)\) is an ideal of \(R\) and the restriction \(\alpha_g\) of \(\beta_g\) to \(S_{g^{-1}}\), with \(g\in G\), is a partial action of \(G\) to \(R\).NEWLINENEWLINE Let \((R,\alpha)\) be a partial action of a group \(G\) on \(R\). An ideal \(I\) of \(R\) is said to be \(\alpha\)-invariant if \(\alpha_g(I\cap S_{g^{-1}})=I\cap S_g\) for every \(g\in G\). The ring \(R\) is said to be \(\alpha\)-prime if for every nonzero \(\alpha\)-invariant ideals \(A,B\) of \(R\), \(AB\neq 0\) and \(\alpha\)-semiprime if for every nonzero \(\alpha\)-invariant ideal \(A\) of \(R\), \(A^2\neq 0\).NEWLINENEWLINE The Martindale ring of \(\alpha\)-quotients, \(Q_m(R)\), of an \(\alpha\)-semiprime ring \(R\) can be build as the direct limit of homomorphisms on \(\alpha\)-invariant ideals of zero annihilators (the set of these ideals is a power filter which is denoted by \(F(R)\)) NEWLINE\[NEWLINEQ_m(R)=\varinjlim\Hom(I,R),\quad I\in F(R),NEWLINE\]NEWLINE i.e., \(Q_m(R)\) consists on equivalence classes of partial homomorphisms \([f,I]\) for \(I\in F(R)\), where \([f_1,I_1]=[f_2,I_2]\) if \(f_1\) coincides with \(f_2\) on \(I\in F(R)\) such that \(I\subset I_1\cap I_2\).NEWLINENEWLINE Among other results the author extends the partial action \(\alpha\) of \(R\) to a partial action of \(Q_m(R)\) and relates the enveloping action of the Martindale ring of \(\alpha\)-quotients of \(R\) with the Martindale ring of \(\beta\)-quotients of the enveloping action of \((R,\alpha)\).