So I have to answer this question:
Let U be the set of expressions of the form a + bu where a and b are real numbers. The u is a formal symbol. We want to make U into a ring similar to how we defined C as a ring, and we’d like u to satisfy the equation $u^2 = 2u - 2$. Provide the formulas we should use to define + and · in U.
Anybody understand what is meant by ''provide the formulas...''? Do I simply have to define addition and multiplication in U in the exact same way as in** C**? If not, what am I to do?
If the ring operations have to satisfy the ring axioms, then there is only one way to define addition and multiplication: $$ (a_1+b_1u)+(a_2+b_2u)=a_1+b_1u+a_2+b_2u=a_1+a_2+b_1u+b_2u=(a_1+a_2)+(b_1+b_2)u $$ $$ (a_1+b_1u)\cdot(a_2+b_2u)=(a_1 a_2)+(a_1b_2+a_2b_1)u+b_1b_2u^2=\cdots $$ A subtle point is that you need to require that $u$ commutes with real numbers.
Another take is to consider the matrix representation and use the usual matrix operations: $$ a+bu \leftrightarrow \pmatrix{ a & -2b \\ b & a+2b} $$ This representation comes from the map $z \mapsto (a+bu)z$ in the basis $1,u$.