Are there any references on linear algebra (module theory) over non unital rings?

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Are there any references on linear algebra (module theory) over non unital rings? What are the main differences with unital rings in that respect?

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Maybe not exactly what you're looking for, but these notes of Quillen say more about foundational matters regarding modules over non-unital rings than you probably thought possible!

He argues that the "correct" (I'm paraphrasing: he doesn't actually use that word) modules to consider are what he calls "firm modules".