Rings with projective socle

Abstract

The class of rings with projective left socle is shown to be closed under the formation of polynomial and power series extensions, direct products, and matrix rings. It is proved that a ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a projective left socle if and only if the right annihilator of every maximal left ideal is of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">fR</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an idempotent in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This result is used to establish the closure properties above except for matrix rings. To prove this we characterise the rings of the title by the property of having a faithful module with projective socle, and show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has such a module, then so does <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript n Baseline left-parenthesis upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_n}\left ( R \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In fact we obtain more than Morita invariance. Also an example is given to show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e upper R e"> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mi>R</mml:mi> <mml:mi>e</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">eRe</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for an idempotent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e"> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding="application/x-tex">e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with projective socle, need not have projective socle. The same example shows that the notion is not left-right symmetric.