All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classifications 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 depends on roots of certain system of polynomials at structure constants of the algebra. We also give explicit forms of the polynomials.

中文翻译：

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二维代数的子代数、幂等、理想和准单位

描述了二维代数的所有子代数、幂等、左（右）理想和左准单位。提供了具有给定数量的子代数、左（右）理想的代数分类。特别是，给出了所有简单二维代数的同构类列表。在理想和子代数的研究中，它们的个数取决于代数结构常数处某个多项式系统的根。我们还给出了多项式的显式形式。