There are several connections between algorithms for classical persistent homology, representations of persistence modules, and computational algebra. The connections to matrix algebra and linear algebra are relatively well known, and anchored in how the community works with and talks about algorithms. Far less pervasive, but a good source of intuitions and techniques, are methods originally developed to deal with more complex cases: Gröbner bases build up the core of computational commutative algebra, handling quotient rings of polynomial rings in several variables and their modules. The techniques and terminology from Gröbner basis techniques match up nicely - not only with multi-dimensional persistence, but also with techniques and algorithms for classical persistence - and can contribute intuitions and new algorithms. In this talk, we will be looking into the basics of computational commutative algebra and find points of contact with the classical persistence algorithm. Based on this new language, we will then introduce perspectives on algebraic constructions for persistence modules that generalize and build new contexts for previous work on kernel, cokernel and image persistence.