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**Partial actions and partial skew group rings.**
*(English)*
Zbl 1148.16022

This paper extends several classical results to partial skew group rings. A partial action of a group on an algebra is a collection of ideals \(D_g\) and isomorphisms \(\alpha_g\colon D_g\to D_{g^{-1}}\) satisfying some natural constraints. A partial skew group ring is the set of all finite formal sums \(\sum_ga_gu_g\) with \(a_g\) from the ideal \(D_g\), with addition as usual and multiplication induced by \((a_gu_g)(a_hu_h)=\alpha_g(\alpha_{g^{-1}}(a_g)a_h)u_{gh}\). These notions were introduced by M. Dokuchaev and R. Exel [Trans. Am. Math. Soc. 357, No. 5, 1931-1952 (2005; Zbl 1072.16025)].

After proving some general results on partial actions of a group on a ring possessing an enveloping action, the authors consider the question of associativity of the (quasi) partial skew group ring (which is not always associative, however, in case there is an enveloping action, then it is indeed associative), and prove that the factor ring of the quasi partial skew group ring by the radical is well-defined and associative. Furthermore, under the assumption that there is an enveloping action, generalizations of classical results such as Maschke’s theorem, results on von Neumann regularity, prime ideals and the prime radical, primitive ideals and the Jacobson radical are studied for partial skew group rings. For instance, if the order of the group is invertible in the ring, then the Jacobson radical of the partial skew group ring is the partial skew group ring of the group over the Jacobson radical.

After proving some general results on partial actions of a group on a ring possessing an enveloping action, the authors consider the question of associativity of the (quasi) partial skew group ring (which is not always associative, however, in case there is an enveloping action, then it is indeed associative), and prove that the factor ring of the quasi partial skew group ring by the radical is well-defined and associative. Furthermore, under the assumption that there is an enveloping action, generalizations of classical results such as Maschke’s theorem, results on von Neumann regularity, prime ideals and the prime radical, primitive ideals and the Jacobson radical are studied for partial skew group rings. For instance, if the order of the group is invertible in the ring, then the Jacobson radical of the partial skew group ring is the partial skew group ring of the group over the Jacobson radical.

Reviewer: János Kurdics (Nyíregyháza)

### MSC:

16S35 | Twisted and skew group rings, crossed products |

16D25 | Ideals in associative algebras |

16N20 | Jacobson radical, quasimultiplication |

16W22 | Actions of groups and semigroups; invariant theory (associative rings and algebras) |

### Keywords:

partial actions; partial skew group rings; associativity; prime ideals; prime radical; primitive ideals; Jacobson radical### Citations:

Zbl 1072.16025
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\textit{M. Ferrero} and \textit{J. Lazzarin}, J. Algebra 319, No. 12, 5247--5264 (2008; Zbl 1148.16022)

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### References:

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