A note on Fuller's theorem (Q581634)

Let R, \(\Delta\) be rings, \({}_ RC\) a complete additive subcategory of R-Mod, T: \(\Delta\)-Mod\(\to_ RC\) a category equivalence, \({}_ RU=T(\Delta)\). If \({}_ RU\) is projective then \({}_ RC\) is closed under extensions and for simple left R-modules S, \(S'\) it is \(S'\in_ RC\) provided \(S\in_ RC\) and \(Ext^ 1_ R(S,S')\neq 0\). The converse holds if R is right perfect and each simple module in \({}_ RC\) is finitely presented (Th.2). If R is left Artinian and \({}_ RC\) is category equivalent to R-Mod, then \({}_ RC=R\)-Mod (Th.5).