Geodesic
Endomorphism rings of free modules
Sbornik. Mathematics, Tome 23 (1974) no. 2, pp. 215-231

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Suppose $\mathfrak a$ is some property of modules. Let $\mathfrak{R_a}$ denote the class of rings over which all modules possess property $\mathfrak a$. The main theorem of this paper answers the following question for a rather extensive class of properties $\mathfrak a$; what must the property $\mathfrak b$ of modules be in order that $R\in\mathfrak{R_a}$ if and only if $\operatorname{End}_R(F)\in\mathfrak{R_b}$, for any free $R$-module $F$? Among the corollaries are many well-known theorems relating properties of the ring $R$ and the rings $\operatorname{End}_R(F)$, and also a number of new results of similar type. Bibliography: 35 titles. 
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     author = {G. M. Brodskii},
     title = {Endomorphism rings of free modules},
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G. M. Brodskii. Endomorphism rings of free modules. Sbornik. Mathematics, Tome 23 (1974) no. 2, pp. 215-231. http://geodesic.mathdoc.fr/item/SM_1974_23_2_a3/