Formal representation of rings in computations While reading a paper about using algebraic methods to detect some induced subgraphs, it appears that edge ideal is an important tool connecting commutative algebra and graph theory. Since I'm not familiar with computations of algebraic objects, is there any good references or books on this topic? Particularity in representing a ring R on a Turing machine, and the complexity of deciding basic properties on R (say, the height of an prime ideal in R.)
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.