Partial actions and partial fixed rings. (English) Zbl 1202.16035

Ávila, Jesús; Ferrero, Miguel; Lazzarin, João

Commun. Algebra 38, No. 6, 2079-2091 (2010).

Let \(k\) be a commutative ring with 1, \(R\) a unital \(k\)-algebra and \(G\) a group. A partial action \(\alpha\) of \(G\) on \(R\) is a collection \(\{(D_g,\alpha_g)\mid g\in G\}\) where \(D_g\) are ideals of \(R\) and \(\alpha_g\colon D_{g^{-1}}\to D_g\) are isomorphisms of \(k\)-algebras such that
(i) \(D_1=R\) and \(\alpha_1\) is the identity map of \(R\);
(ii) \(\alpha_h^{-1}(D_h\cap D_{g^{-1}})\subset D_{(gh)^{-1}}\) for any \(g,h\in G\);
(iii) \(\alpha_g\circ\alpha_h(x)=\alpha_{gh}(x)\) for any \(x\in\alpha_h^{-1}(D_h\cap D_{g^{-1}})\) and \(g,h\in G\).
The partial skew group ring \(R*_\alpha G\) is the set of finite formal sums \(\sum_{g\in G}a_g\delta_g\), \(a_g\in D_g\) for every \(g\in G\) where the addition is defined in the usual way and \((a_g\delta_g)(b_h\delta_h)=\alpha_g(\alpha_{g^{-1}}(a_g)b_h)\delta_{gh}\).
The enveloping algebra \((T,\beta)\) of \(\{(D_g,\alpha_g)\mid g\in G\}\) satisfies (i) \(T=\sum_{\beta\in G}\beta_g(R)\); (ii) \(D_g=R\cap\beta_g(R)\) for each \(g\in G\); (iii) \(\alpha_g(x)=\beta_g(x)\) for all \(g\in G\) and \(x\in D_{g^{-1}}\).
Let \(G\) be finite. Then \(D_g=R1_g\) for a central idempotent \(1_g\) of \(R\) and the subring \(R^\alpha\) of the elements of \(R\) fixed under \(\alpha\) is \(\{x\in R\mid\alpha_g(x1_g)=x1_g\) for any \(g\in G\}\), and \(\text{tr}_\alpha\colon R\to R^\alpha\) is given by \(\text{tr}_\alpha(r)=\sum_{\alpha\in G}\alpha_g(r1_{g^{-1}})\) for any \(r\in R\) and \(\text{tr}_G(t)=\sum_{g\in G}\beta_g(t)\) for any \(t\in T\). It is shown that \(R^\alpha\cong T^G\), and this derives generalizations of several well known results for global actions for the prime and Jacobson radicals \(\text{Nil}(R)\), \(\text{Jac}(R)\).
Theorem. (1) Assume \(R\) is \(|G|\)-torsion free. Then \(\text{Nil}(R^\alpha)=\text{Nil}(R)\cap R^\alpha\). (2) Assume \(|G|^{-1}\in R\). Then \(\text{Jac}(R^\alpha)=\text{Jac}(R)\cap R^\alpha\).
Let \(S=R*_\alpha G\). Then some properties of the elements \(\sum_{g\in G}1_g\delta_g\) and \(\text{tr}_\alpha(1_R)^{-1}(\sum_{g\in G}1_g\delta_g)\) are given for an invertible \(\text{tr}_\alpha(1_R)\), and it is shown that if \(R\) is semisimple (resp. semiprimitive, left Artinian, left Noetherian, von Neumann regular), then so is \(R^\alpha\).
Moreover, let \(R\) be a semiprime ring and \(\text{udim}(R)\) the left uniform dimension of \(R\). Conditions are given for \(R^\alpha\) to be a left Goldie ring, and it is shown that \(\text{udim}(R^\alpha)\leq\text{udim}(R)\leq|G|\text{udim}(R^\alpha)\) when \(R\) is \(|G|\)-torsion free.

Reviewer: George Szeto (Peoria)

MSC:

16W22	Actions of groups and semigroups; invariant theory (associative rings and algebras)
16S35	Twisted and skew group rings, crossed products
16N80	General radicals and associative rings

Keywords:

enveloping actions; fixed subrings; partial actions; prime radical; Jacobson radical; semisimplicity; partial skew group rings

References:

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