Artinian skew group rings

Abstract

Let <italic>R</italic> be a ring with identity and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi>θ<!-- θ --></mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group homomorphism from a group <italic>G</italic> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t left-parenthesis upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>Aut</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\operatorname {Aut}}(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the group of automorphisms of <italic>R</italic>. We prove that skew group ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R asterisk Subscript theta Baseline upper G"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo>∗<!-- ∗ --></mml:mo> <mml:mi>θ<!-- θ --></mml:mi> </mml:msub> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">R{ \ast _\theta }G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is right Artinian (resp., semiprimary, right perfect) if and only if <italic>R</italic> is right Artinian (resp., semiprimary, right perfect) and the group <italic>G</italic> is finite. Also semilocal skew group rings over fields are characterized.

Locations

• Proceedings of the American Mathematical Society - View - PDF