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.