I'm studying Morita theory of equivalence of categories of modules, putting together different books and notes and trying to develop it in a clear way.

1)I know the proof of this fact: Let $R$ be a ring and let $M_R$ be a generator. Let $A=End(M_R)$, then the bimodule ${}_{A}M_R$ is faithfully balanced. This proof uses the following lemma: for a ring $R$ and a right $R-$module $M$, for every $m \in M$ and every $ f \in Hom_R(M,R)$ the map $m \cdot f \in End_R(M)$, where $(m \cdot f)(x)=m\cdot (f(x))$.

I've tried to rewrite this for left modules because next I'll use this, and I think I've done right, but if someone can help me, I'll write my left version to be sure it's okay.

2)I have to prove this: Let $R$ and $S$ be equivalent rings via $F:R-$mod $\to S-mod$ and $G$ the inverse functor. Let $P=F({}_{R}R)$ and $Q=G({}_{S}S)$. Then ${}_{R}P_S$ and ${}_{S}Q_R$ are faithfully balanced bimodules and ${}_{S}P, P_R, {}_{R}Q, Q_S$ are progenerators.

To do this, I've followed my book: there is an isomorphism between $R$ and $End({}_{S}P)$ so I can give $P$ a right $R$-module structure and ${}_{S}P_R$ is a bimodule. Then ${}_{R}R$ is a progenerator and so is ${}_{S}P$ (I know that being generator, projective and finitely generated is a Morita invariant, i.e. a property preserved via equivalence of categories). Now my book uses a lot of properties proved before, but I have to develop this argument in few pages, so I used the property 1) (in the left version) to say that ${}_{S}P_{End({}_{S}P)}$ is a faithfully balanced bimodule and so is ${}_{S}P_R$ because $R$ with his natural action on ${}_{S}P$ is isomorphic to $End({}_{S}P)$. Now I'd like to say that $P_R$ is a progenerator. Again my book uses another property, but I've found this one:

3)if ${}_{S}P_R$ is a faithfully balanced bimodule the following are equivalent: ${}_{S}P$ is a progenerator; $P_R$ is a progenerator. (I want to be sure I've understood: when I see ${}_{S}P$ progenerator it means progenerator of the category of left $S$-modules and $P_R$ for the one of right $R$-module?) (I have some doubts about the proof, but I'll think about it later) So I thought of using this one. Is it all right since there?

The properties used by my book to prove point 2) are the following two.

a) Let ${}_{S}P_R$ bimodule faithfully balanced. Then ${}_{S}P$ is generator if and only if $P_R$ is finitely generated and projective.

b) a left $R-$ module ${}_{R}G$ is a generator if and only if ${}_{R}G$ is faithful and balanced and $G_{End({}_{R}G)}$ is finitely generated and projective. (Here ${}_{R}G$ balanced means that ${}_{R}G_{End({}_{R}G)}$ is a balanced bimodule).

I'm not sure about using this two properties to prove fact 2) because for this 2 proofs the book uses a lot of facts in the previous chapter and I can't write too much for the paper I have to do. Is it ok the way I've done (without properties a) and b))? Or are there proofs of a) and b) which are short?

I'm also confused because then I'll have to prove also this result: (after proving that left categories $R-$mod and $S-$mod are equivalent if and only if the right ones are)

4)Let $R$ and $S$ rings, the following are equivalent:

$R$ and $S$ are equivalent; There is a progenerator $P_R$ with $S$ and $End(P_R)$ isomorphic; There is a progenerator ${}_{R}Q$ with $S$ and $End({}_{R}Q)$ isomorphic

To prove this, my book again uses properties a) and b). If I decide not to use it, can I prove this fact using only fact 1) and 3)?

Sorry if the question is very long but I'm really confused and I'd like to make this situation clearer in my mind, so any help will be appreciated! Thanks!


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