Left and right projections are equivalent in Rickart \(C^*\)-algebras (Q909201)

\textit{I. Kaplansky} [Ann. Math., II. Ser. 53, 235-249 (1951; Zbl 0042.124)] conjectured that in every Rickert \(C^*\)-algebra left projections are equivalent to right projections. For finite Rickart \(C^*\)-algebras this was proved by \textit{D. Handelman} [Finite Rickart \(C^*\)-algebras and their properties, Adv. Math. Suppl. Studies 4, 171-196 (1979; Zbl 0511.46054)]. In the present paper the author constructs for any Rickart \(C^*\)- algebra A a closed ideal I(A) such that A/I(A) is a finite Rickart \(C^*\)-algebra and I(A) is contained in every ideal K such that A/K is finite. With the help of this and Handelman's result, the author proves the conjecture of Kaplansky in the general case.