I have a question:
Describe up to isomorphism all 2d associative algebras over Ring with 1 and without 1
That's all info, that I found about this question:
Every two-dimensional associative algebra is either equivalent or isomorphic to one of the eight algebras.
But I don't know how it can be proved. There was something similar to my problem, but with Lie-group theory which I don't understand.
The notion "2d" makes sense for algebras over a field (namely the dimension of its underlying $K$-vecor space), but is not "defined" over rings. I suppose you mean $R$ perhaps as $\mathbb{R}$, the field of real numbers. Goze and Remm have classified all $2$-dimensional algebras over an arbitrary field $K$, see here. It is easy to pick out the associative ones form their list. Independently, many people already did this classification, see for example the recent colourful slides here. The idea is easy. Take as basis $(e_1,e_2)$ and write all basis products $e_i.e_j$ in this basis, with coefficients $a_{ij}$. Then determine the conditions for associativity and compute isomorphisms.
