Let M={0,1,a,b} be a commutative ring with unity. I'm supposedt to fill in the Cayley tables. Can someone help me with the table for the multiply table. I think that $a*b=b*a=a$ do to commutative ring. Is the multiplicative identity $1$? Is $0*1=0$ for example?
I think that the Cayley table for + is:
The elements of a commutative ring under addition must be an abelian group. Since there are four elements, that group in this case must be $\mathbb Z_4$ or $\mathbb Z_2^2$. Since there is a non-zero element ($1$) for which $1+1\ne0$, the abelian group must be $\mathbb Z_4$ and the ring is isomorphic to $\mathbb Z/4\mathbb Z$ with the associations $a=2$ and $b=3$. Filling out the tables should now be easy.