General method for defining non-commutative finite associative algebras of arbitrary dimension m ≥ 2 is discussed. General formulas describing local unit elements (the right-, left-, and bi-side ones), square roots of zero and zero divisors are derived. For arbitrary value m the single bi-side unit corresponds to every element of the algebra, except the square roots from zero. Various modifications of the multiplication operation can be assigned using different sets of the values of structural coefficients. It is proved that all of the modifications are mutually associative..