By an FCR-algebra the authors understand an algebra \(A\) over a field \(k\) such that I: Every finite dimensional representation of \(A\) is completely reducible and II: \(A\) is residually finite-dimensional. Examples are the envelopes of semisimple Lie algebras in characteristic \(0\) and the quantum enveloping algebras \({\mathfrak U}_q({\mathfrak g})\), when \(q\) is not a root of \(1\). Other examples have been found recently [\textit{H. Kraft} and \textit{L. W. Small}, Math. Res. Lett. 1, No. 3, 297--307 (1994; Zbl 0849.16036), \textit{I. M. Musson} and \textit{M. Van den Bergh}, Mem. Am. Math. Soc. 650 (1998; Zbl 0928.16019)]. The authors' aim is to give a simple characterization and to apply it. They note that the class of FCR-algebras admits direct products (as long as \(k\) is not too small) and direct sums. Next they prove that for any \(k\)-algebra \(A\) the following are equivalent: (a) \(A\) satisfies I above, (b) every ideal \(I\) of \(A\) is idempotent, (c) if \(M_1\), \(M_2\) are maximal ideals of finite codimension in \(A\), then \(M_1\cap M_2= M_1 M_2= M_2 M_1\). As a consequence the class of \(C^*\)-algebras and the class of von Neumann regular rings both satisfy condition I. This condition is also verified for any localization of an FCR-algebra at a right Ore set.