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

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.

### 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
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### References:

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