Half-orthogonal sets of idempotents. (Q2796512)

The purpose of this paper is to improve several results on lifting idempotents.NEWLINENEWLINE Throughout, \(R\) denotes an associative ring with \(1\), \(J(R)\) the Jacobson radical of \(R\), \(U(R)\) the group of units of \(R\), and \(\mathrm{idem}(R)\) the set of idempotents of \(R\).NEWLINENEWLINE We note some of the more important results:NEWLINENEWLINE An ideal \(I\) of \(R\) is called \textit{enabling} if given \(x\in R\) and \(e\in\mathrm{idem}(R)\) with \(x\equiv e\pmod I\), then there exists an idempotent \(f^2=f\in Rx\) with \(f\equiv e\pmod I\). Now, let \(a,b\in R\), \(e\in\mathrm{idem}(R)\), and let \(I\) be an enabling ideal of \(R\). If \(ab\equiv e\pmod I\), then there exists \(f\in\mathrm{idem}(R)\) with \(f\in bRa\) and \(f\equiv bea\pmod I\). If it is additionally assumed that \(a\in\mathrm{idem}(R)\) with \(a\equiv ea\pmod I\) and \(I\subseteq J(R)\), then \(Ra=Rf\) (i.e., \(f\) and \(a\) are \textit{left associates}). As a corollary, if \(x\in R\), \(e\in\mathrm{idem}(R)\), \(I\subseteq J(R)\) and \(exe\equiv e\pmod I\), then there exists \(f\in\mathrm{idem}(R)\) with \(f\in xRx\), \(Rf=Re\) and \(f\equiv xe\pmod I\).NEWLINENEWLINE Assume that \(e_1,e_2,\ldots,e_n\in\mathrm{idem}(R)\) such that \(e_ie_j\in J(R)\) whenever \(i<j\). Then there exists \(u\in U(R)\) for which \(\{ue_i\}\) and \(\{e_iu\}\) are orthogonal families of idempotents. This result can be extended to the infinite case: If \(\Gamma\) is a totally ordered set and \(\{e_i\}_{i\in\Gamma}\subseteq\mathrm{idem}(R)\) where \(e_ie_j\in J(R)\) whenever \(i<j\), then the left ideal \(\sum_{i\in\Gamma}Re_i\) is a direct sum, and the family \(\{Re_i\}_{i\in\Gamma}\) is a local direct summand of \(_RR\) (i.e., \(\sum_{i\in F}Re_i\) is a direct summand for each finite subset \(F\subseteq\Gamma\)). Furthermore, \(\sum_{i\in \Gamma}Re_i\) is a direct summand of \(_RR\) if and only if \(e_i=0\) for all but finitely many \(i\in\Gamma\). (To overcome the obstacle of interpreting infinite sums of idempotents, \(R\) is endowed with a (left linear) Hausdorff topology, which is, under suitable conditions, \(\Sigma\)-complete.)NEWLINENEWLINE The final section sheds some light on so-called Harada modules.