Mathematics & Physics 2016, 9(4), 443–448 УДК 512.64+512.55 The Determinants over Associative Rings: a Definition, Properties, New Formulas and a Computational Complexity Georgy P. Egorychev∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 17.06.2016, received in revised form 05.07.2016, accepted 15.09.2016 We give a new definition for the determinants over an associative ring Q and study their properties. <...> In particular, we obtain a new family of polynomial identities (computational formulas) for these determinants that contain up to n! free variables. <...> Introduction Recently the author in [1,2] has found a new family of formulas with free variables (polynomial identities) for the determinant over the commutative ring K. These formulas have alloved me to give a new definition for determinants (an e-determinant edet(A)) over a wide class of rings [1,3] and to establish their basic properties (without proof). <...> These definitions (for a noncommutative case) are closely related to a definition of the symmetrized Barvinok determinant sdet(A) [4]. <...> Here the complete proof of statements from [3] for edet(A) over the associative ring Q is given. <...> With the help of knoun theorem of polarization [5,6] to obtain a new family of polynomial identities for edet(A) that contain up to n! free variables. <...> These propeties coincide with many well-known propeties of a determinant over a commutative ring. <...> We also estimate a computational complexity of the obtained formulas for determinants. <...> Some prospects of the basic results are specified. 1. <...> Notations and definitions Let K be a commutative ring, K a noncommutative associative ring, Q a noncommutative ring with associative n-powers (one-monomial associativeness), and Q be an associative ring; let also each ring be with division by integers. <...> All rights reserved c – 443 – n be the subsets of even and odd permutations Georgy P. Egorychev The Determinants over Associative Rings: a Definition, Properties . . . of length k of matrix A. We denote by L(e) diagonals from S(e) n (S(o) det(A) := ∑ σ∈Sn n ), k = 1, . . . ,n. <...> A classic definition for the determinant over the commutative ring K: (−1)τ(σ)a1σ(1) . . . anσ(n) = 1 ∑ ∑(−1)τ(µ)+τ(σ)aµ(1)σ(1) . . . aµ(n <...>