On separable extensions of group rings and quaternion rings release_xdbbak4owjgl5n47ttq4sl6q2m

Abstract

The purposes of the present paper are (1) to give a necessary and sufficient condition for the uniqueness of the separable idempotent for a separable group ring extension<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$RG$"><mml:mrow><mml:mi>R</mml:mi><mml:mi>G</mml:mi></mml:mrow></mml:math>(<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$R$"><mml:mi>R</mml:mi></mml:math>may be a non-commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extension<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$RQ$"><mml:mrow><mml:mi>R</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:math>over a ring<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$R$"><mml:mi>R</mml:mi></mml:math>, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$Q$"><mml:mi>Q</mml:mi></mml:math>are the usual quaternions<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$i,j,k$"><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math>and multiplication and addition are defined as quaternion algebras over a field. We shall show that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$RG$"><mml:mrow><mml:mi>R</mml:mi><mml:mi>G</mml:mi></mml:mrow></mml:math>has a unique separable idempotent if and only if<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$G$"><mml:mi>G</mml:mi></mml:math>is abelian, that there are more than one separable idempotents for a separable quaternion ring<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$RQ$"><mml:mrow><mml:mi>R</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:math>, and that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$RQ$"><mml:mrow><mml:mi>R</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:math>is separable if and only if<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$2$"><mml:mi>2</mml:mi></mml:math>is invertible in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$R$"><mml:mi>R</mml:mi></mml:math>.
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