One of the best concise guides I've encountered to the relationship between the various algebraic operations -- and how they circle back to one another -- is chapter 22 "Algebra" of volume I of the Feynman Lectures on Physics. For some reason this tied everything together in my head in a way that made so much sense, when the pedagogy I head leading up to it simply did not supply me the same vantage point.
From there I guess I've always thought as the various algebraic operations as functions which build upon one another. I think the article's point about not overlooking multiplication's Dimensionality as a key component is a good one, but neither should that focus avoid talking about the functional relationships of the various operations.
https://www.feynmanlectures.caltech.edu/I_22.html
From there I guess I've always thought as the various algebraic operations as functions which build upon one another. I think the article's point about not overlooking multiplication's Dimensionality as a key component is a good one, but neither should that focus avoid talking about the functional relationships of the various operations.