rt.representation theory - What is the abstract relationship between an indecomposable representation and a sum of irreducible representations with the same character?

$mathbb{Z}$ is the simplest example of a group with indecomposable representations which are not irreducible. Over $mathbb{C}$, the isomorphism classes of $n$-dimensional representations of $mathbb{Z}$ are precisely the conjugacy classes of elements of $GL_n(mathbb{C})$; in particular the indecomposable ones are given by Jordan blocks. The representation corresponding to a Jordan block of size $n$ with diagonal entries $lambda$ has the same character as, but is not isomorphic to, the representation corresponding to a diagonal matrix with entries $lambda$. What is an abstract way to describe this relationship that does not refer to characters? (I am mostly interested in how to describe the relationship between an indecomposable representation and a sum of one-dimensional representations with the same character.)

Motivation

A natural way to study an (associative, unital) algebra $A$ over $mathbb{C}$ (to fix ideas) is to study the category $text{Rep}(A)$ of, say, finite-dimensional representations of $A$. However, if $A$ happens to be commutative and Noetherian, then we do something different: we privilege the one-dimensional representations and call them points, and then we analyze higher-dimensional representations as certain structures on the points. What abstract relationship, from the representation-theoretic perspective, between the one-dimensional and higher-dimensional representations lets us do this?