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All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classification of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of isomorphism classes of all simple two-dimensional algebras is given. In the study of ideals and subalgebras the number of them depend on roots of certain system of polynomials at structure constants of the algebra. We also give explicit forms of the polynomials.

Subalgebras, idempotents, ideals and quasi-units of two-dimensional algebras