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Inspired by Yang Zhang's talk last week (abstract) on the problems of solving complicated matrix equations, in particular variants on the Sylvester equation, I thought it was a good time to review the purely linear context.

If we want to talk about systems of linear equatiosn, what is the basic minimum algebraic context? We need to talk about linear combinations of variables: addition, multiplication, so we need a ring \(R\). The ring elements have to act on the solutions additively, so the solutions must at the very least be in an abelian group: we need a \(R\)-module.

There are then two basic problems: How do we decide if a system is solvable? How do we find the solutions if it is? The first problem has essentially the same answer as in a first course in linear algebra; but this leads us through an unexpected pathway to an intermediate question: Where do we find the solutions?

Along the way we discover that systems of infinitely many equations in infinitely many variables are important, and we learn about injective and pure-injective modules.