Formal representation of rings in computations
Your questions are related to a field (no pun intended) called "Computer Algebra". I myself was looking for comprehensive surveys when I was working on algebraic methods to calculate various graph centrality metrics. I could not find good surveys, but this book was partially helpful. The research papers on this "topic" are scattered all over and often not explicitly categorized as "computer algebra". Reading algorithmic papers on isomorphism, factoring (integers/polynomials) and graph algorithms based on matrix multiplication might give you more insights.