Lifting idempotents and exchange rings

Abstract

Idempotents can be lifted modulo a one-sided ideal <italic>L</italic> of a ring <italic>R</italic> if, given <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper R"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \in R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x minus x squared element-of upper L"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x - {x^2} \in L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists an idempotent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e element-of upper R"> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">e \in R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e minus x element-of upper L"> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">e - x \in L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module <italic>P</italic> has the finite exchange property if and only if, whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P equals upper N plus upper M"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">P = N + M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <italic>N</italic> and <italic>M</italic> are submodules, there is a decomposition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P equals upper A circled-plus upper B"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mo>⊕</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">P = A \oplus B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of-or-equal-to upper N"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A \subseteq N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B subset-of-or-equal-to upper M"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">B \subseteq M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.