Let F be a field of characteristic zero and let A be a twodimensional non-associative algebra over F. We prove that the sequence cn( A), n= 1, 2,..., of codimensions of A is either bounded by n + 1 or grows exponentially as 2(n). We also construct a family of two-dimensional algebras indexed by rational numbers with distinct T-ideals of polynomial identities and whose codimension sequence is n + 1, n >= 2.
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