  Extending the thoroughly studied theory of group stability, we study Ulam
stability type problems for associative and Lie algebras; namely, we
investigate obstacles to rank-approximation of almost solutions by exact
solutions for systems of polynomial equations. This leads to a rich theory of
stable associative and Lie algebras, with connections to linear soficity,
amenability, growth, and group stability. We develop rank-stability and
instability tests, examine the effect of algebraic constructions on
rank-stability, and prove that while finite-dimensional associative algebras
are rank-stable, `most' finite-dimensional Lie algebras are not.
