For associative algebras, as your required, see Plotkin, Algebras with the same (algebraic) geometry, Israel J. Math., 96 (2) (1996), 511–522.
This is, being more precise, part of this nice relatively new field of Universal Algebraic Geometry which discuss such things.
For a survey I recommend A. Shevlyakov, Lectures notes in universal algebraic geometry, https://arxiv.org/abs/1601.02743.
EDIT depending on the kind of question you are interested in, pherhaps the theory of (associative) PI-algebras might be also interesting. For this I recommend V. Drensky's Free Algebras and PI-algebras.