I've encountered the following definition, which I'm struggling to get my head around:
Let $(A, \mu, \eta, \epsilon)$ be an augmented algebra, with augmentation ideal $I(A) = \ker(\epsilon)$. We define the module of indecomposables to be $QA = I(A)/I(A)I(A)$.
What is the best way to interpret this definition? It appears that we want to impose the condition that all non-trivial products become trivial. If this is the case however, why do we not define $QA = A/I(A)I(A)$?
Further, does the module of indecomposables have any relation to indecomposable modules?
Thanks!
Yes, it's exactly the way you said it: you annihilate all non-trivial products. And indeed, you could also consider $A/I(A)^2$; this is related to $I(A)/I(A)^2$ through a split short exact sequence of ${\mathbb k}$-modules $$0\to I(A)/I(A)^2\longrightarrow A/I(A)^2\longrightarrow A/I(A)\cong {\mathbb k}\to 0,$$ showing that $A/I(A)^2\cong I(A)/I(A)^2\oplus {\mathbb k}$ as ${\mathbb k}$-modules.
I'm not aware of any connection with indecomposable modules.