So, I'm attempting to work through Silting modules by Hügel, Marks, Vitória and I'm having some difficulties with certain concepts and notation. I will "present" what I think these concepts are, but would very much like to be corrected if I've misunderstood them.
Let $M\in \operatorname{Mod}(R)$ for a unitary ring $R$.
$\operatorname{Add}(M)$ denotes the additive closure of $M$ consisting of all modules isomorphic to a direct summand of an (arbitrary) direct sum of copies of $M$, while $\operatorname{Gen}(M)$ is the subcategory of $M$-generated modules (that is, all epimorphic images of modules in $\operatorname{Add}(M)$
First, I would like to understand exactly what $\operatorname{Add}(M)$ looks like. If $A\in \operatorname{Add}(M)$, does this simply mean that $A\cong M^{k}$ for some $k\in \mathbb{N}$, as $M^{k}\oplus M$ is an arbitrary direct sum of copies of $M$?
Secondly, if $G\in \operatorname{Gen}(M)$, is it then true that $G\cong Im(\phi)$ for some epimorphism $\phi : A \rightarrow G$ where $A\cong M^{k}$ for some $k\in \mathbb{N}$?
Finally, regarding $^\circ(M^\circ)$, here is the definition of $M^\circ$:
... we denote by $M^\circ$ the subcategory of $\operatorname{Mod}(R)$ consisting of the objects $N$ such that $\operatorname{Hom}_R(M,N)=0$
There is no definition of $^\circ(M^\circ)$, and I can only assume that it's supposed to be "clear to the reader". Initially I thought it might be $$^\circ(M^\circ):= \text{subcategory of } \operatorname{Mod}(R) \text{ consisting of } K \text{ such that } \operatorname{Hom}_R(K,N) = 0 \text{ for all } N\in M^\circ$$ but that is a "wild guess" at best.
I would greatly appreciate if someone familiar with these concepts would explain them to me!