Endomorphism rings of free modules

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.
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