In the literature on representation theory of finite dimensional algebras, a left (and similarly right) minimal homomorphism is defined as the following:
For a pair of modules $L $ and $M$ in $mod-A$, an $A-$homomorphism $f:L\rightarrow M$ is called left minimal if for every $h\in End(M)$, if $h\circ f = f$, then $h\in Aut(M)$.
Here are my questions:
- Since for any arbitrary module $X$ in $mod-A$, we can consider the set of all pairs ($Y, f)$, where $f:Y\rightarrow X$ is a left minimal homomorphism, and this set is always nonempty (it at least contains $(X,1_X)$ ), could we possibly use inclusion as a partial order and talk about the smallest $(Y,f)$, which may not be necessarily unique, as the best object with such a property? Moreover, if two objects themselves are not comparable (with regard to the partial order defined by inclusion) their images in $X$ might be. So, we could possibly compare the images as well and talk about something like "best left minimal homomorphism(s) for a given module $X$ in $mod-A$".
Does it make sense at all to think about such a notion?
- So far, I have seen the aforementioned notions (Left and right minimal homomorphisms) only in the literature of representation of finite dimensional algebras. But, at least at this step, the definitions seem to be independent of the good properties that this category possesses. Could one study these type of morphisms in any other category (at least abelian categories). If so, which seems to be plausible, I was wondering if you could let me know why they only (as long as I know) appear in this area.
I already know that they are used to firstly define left and right minimal almost split morphisms and consequently almost split sequences, which lead to the proof of the first Brauer_Thrall Conjecture. But, I am curious to know if they also have other applications in other areas.