Artinian (Noetherian) part of a Goldie ring (Q1823296)

The author studies the right (left) artinian radical (i.e. the sum of all right (left) ideals which are artinian as right (left) R-modules) of a Goldie ring R and obtains some results similar to those of the case when R is noetherian. However, if the author paid attention to the fact that a semiprime right Goldie, right quotient ring is semisimple artinian, then he would see from Lemma 2.1.2 that the statements of Theorem 3.2 and Theorem 3.4 are well-known because rings appearing in these theorems are in fact semisimple artinian. Moreover one can prove a more general result from which it follows that a right Goldie ring satisfying DCC (ACC) on essential right ideals is right artinian (right noetherian): If R is a ring satisfying DCC (ACC) on essential right ideals, then the factor ring of R by its right socle is right artinian (right noetherian). See Proposition 1.1 of \textit{E. P. Armendariz} [Commun. Algebra 8, 299-308 (1980; Zbl 0444.16015)] (resp. Lemma 2 of \textit{Nguyen V. Dung}, \textit{R. Wisbauer} and the reviewer [Arch. Math. 53, 252-255 (1989; Zbl 0658.16018)]).