Linear Algebra and its Applications, Transformations Preserving Matrix Invariants, Algebraic Combinatorics, Max-algebras and their Applications, Noncommutative Rings, PI-Algebras.
Main publications:
A. E. Guterman, Monotone additive transformations on matrices, Mathematical Notes, 81(5), 2007, 681–692.
I. I. Bogdanov, A. E. Guterman, Monotone matrix transformations defined by the group inverse and simultaneous diagonalizability, Matematicheskii Sbornik, 198(1), 2007, 3–20.
L. B. Beasley, A. E. Guterman, K.-T. Kang, S.-Z. Song, Idempotent Boolean matrices and majorization, Fundamental and Applied Mathematics, 13(1), 2007, 11–29.
L. B. Beasley, A. E. Guterman, S.-G. Lee, S.-Z. Song, Frobenius and Dieudonne theorems over semirings, Linear and Multilinear Algebra, 55(1), 2007, 19–34.
L. B. Beasley, A. E. Guterman, Y.-B. Jun, S.-Z. Song, Linear preservers of extremes of rank inequalities over semirings: the row and column ranks, Linear Algebra Appl., 413, 2006, 495–509.
A. A. Alieva, A. E. Guterman, B. Kuzma, Rank-permutable additive mappings, Linear Algebra Appl., 414, 2006, 607–616.
L. B. Beasley, A. E. Guterman, C. L. Neal, Linear preservers for Sylvester and Frobenius bounds on matrix rank, Rocky Mountain Journal of Mathematics, 86(1), 2006, 67–80.
L. B. Beasley, A. E. Guterman, S.-G. Lee, S.-Z. Song, Linear preservers of zeros of matrix polynomials, Linear Algebra and its Applications, 401, 2005, 325–340.
A. A. Alieva, A. E. Guterman, Monotone linear transformations on matrices are invertible, Communications in Algebra, 33, 2005, 3335–3352.
L. B. Beasley, A. E. Guterman, Linear Preservers of Extremes of Rank Inequalities over Semirings: The Factor Rank, J. of Math. Sciences, 131(5), 2005, 5919–5938.
A. A. Alieva, A. E. Guterman, Linear preservers of rank permutability, Linear Algebra and its Applications, 384, 2004, 97–108.
L. B. Beasley, A. E. Guterman, S.-G. Lee, S.-Z. Song, Linear transformations preserving the Grassmannian over $M_n({\bf Z}+)$, Linear Algebra and its Applications, 393, 2004, 39–46.
A. A. Alieva, A. E. Guterman, Rank permutability and additive operators preserving some rank product conditions, Fundamental and Applied Mathematics, 10(4), 2004, 3–14.
L. B. Beasley, A. E. Guterman, S.-C. Yi , Linear preservers of extremes of rank inequalities over semirings: term rank and zero-term rank, Fundamental and Applied Mathematics, 10(2), 2004, 3–21.
A. E. Guterman, A.V. Mikhalev, General algebra and linear transformations preserving matrix invariants, Fundamental and Applied Mathematics, 9(1), 2003, 83–101.
A. E. Guterman, Transformations of non-negative integer matrices preserving the determinant, Uspehi Mat. Nauk, 58(6), 2003, 147–148.
A. A. Alieva, A. E. Guterman, Linear transformations preserving rank-commutativity, Bull. of the Moscow State University. Ser. 1. Mathematics. Mechanics, 2003, no. 6, 11–17.
A. E. Guterman, Monotone Matrix Maps Preserve Non-Maximal Rank, Combinatorial Methods in Polynomial Identities, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 2003, 311–328.
A. E. Guterman, Linear transformations that preserve the Dieudonne determinant over an arbitrary division ring, Uspehi Mat. Nauk, 57(4), 2002, 171–172.
A. Guterman, Linear preservers for matrix inequalities and partial orderings, Linear Algebra and Applications, 331, 2001, 75–87.
A. E. Guterman, Frobenius type theorems in the noncommutative case, Linear and Multilinear Algebra, 48(4), 2001, 293–312.
A. E. Guterman, Identities of near-triangular matrices, Matem. Sbornik, 192(6), 2001, 3–15.
A. Guterman, Linear preservers for Drazin partial order, Communications in Algebra, 29(9), 2001, 3905–3917.
A. Guterman, C.-K. Li, P. Semrl, Some general techniques on Linear Preserver Problems, Linear Algebra and Applications, 315, 2000, 61–81.
A. Guterman, Singularity preservers over local domains, Journal of Mathematical Sciences, 102(6), 2000, 4591–4597.
A. Regev, A. Guterman, On the Growth of Identities, Algebra. Berlin, New York: Walter de Gruyter, 2000, 319–331.