ring homomorphism from $\mathbb Z\oplus\mathbb Z$ into $\mathbb Z\oplus\mathbb Z$.

Question

A question from Gallian which I had in my exam says:

Suppose $\phi$ is a ring homomorphism from $\mathbb Z\oplus\mathbb Z$ into $\mathbb Z\oplus\mathbb Z$.What are the possibilities for $\phi((1,0))?$

Any idea,How to go with this?Please help....

Answer 1

Well, $(1,0)\cdot(1,0)=(1,0)$, so it is an idempotent element (solution of $x^2=x$). Therefore it has to go to some element with the same property. As the product structure on the direct sum is given component-wise, every component of an idempotent element is idempotent in its corresponding factor… i.e. $(a,b)$ is idempotent iff $a^2=a$ and $b^2=b$, so from this you restrict your possibilities to $(1,0)$, $(0,1)$, $(1,1)$, $(0,0)$. Obviously, the first and the last are possible (the identity homomorphism and the null homomorphism). Also the second is just $(a,b)\mapsto (b,a)$. Now you just have to decide if $(1,1)$ is possible…

Answer 2

Note that $(1,0)\cdot (1,0) = (1,0)$ so $\phi((1,0))\cdot \phi((1,0)) = \phi((1,0))$. The only idempotents in $\mathbb{Z}\oplus \mathbb{Z}$ are $(1,0), (0,1), (0,0), (1,1)$. The first three are certainly possibilities. How about $(1,1)$?