The task is to give the description of all (with the precision of isomorphism) commutative rings with unity of $8$ elements.
As for addition definition, I understood that the are two options for the sum of the units: $1+1=0$ and $8 = 0$, for if $n = 0$ we will get a vector space over $\mathbb{F}_n$, and its amount of elements in should be the power of $n$, which is possible only for $2$ and $8$: $2^3$ and $8^1$. In both cases addition is completely defined.
In the case of $8 = 0$ the ring is isomorphic to $\mathbb{Z}/(8)$.
In the second case we can consider the ring as vector space over $\mathbb{F}_2$ of dimension $3$, with elements $\{1, 0, a, a+1,b,b+1,a+b,a+b+1\}$. Now the structure of the ring depends on the definition of multiplication in it. The only fact that I understood about it is that if we define $a^2=a$ and $b^2=b$ then for any element $r$ of the ring $r^2=r$, hence it is boolean ring, which in turn is isomorphic to $\mathbb{F}_2 \times \mathbb{F}_2 \times \mathbb{F}_2$.
Also I know that if we define $a^2=a+1$, then $\{1, 0, a, a+1\}$ is a field of $4$ elements, but how is it connected with the structure of the complete ring?
What are the other options for multiplication and what do they mean in terms of the ring structure?