find all ring homomorphisms from $\Bbb Z\oplus \Bbb Z$ into $\Bbb Z\oplus\Bbb Z$?

Question

As I have tried I found seven homomorphisms as follows:

$f(a,b)=(a,b)$

$f(a,b)=(0,0)$

$f(a,b)=(a,0)$

$f(a,b)=(0,b)$

$f(a,b)=(b,a)$

$f(a,b)=(0,a)$

$f(a,b)=(b,0)$

are there any other possible homomorphisms?

Answer 1

Recall the any ring homomorphism is also a morphism of abelian groups - that is $\mathbf Z$-modules. Each $\mathbf Z$-linear map $\def\Z{\mathbf Z}\Z^2 \to \Z^2$ is represented by a $2\times 2$-matrix $A \in M_{2,2}(\Z)$. Let's check which matrices represent an also multiplicative map: Let $\cdot$ denote multiplication in $\Z^2$, we have \begin{align*} A(e_1\cdot e_1) &= Ae_1 = \binom{a_{11}}{a_{21}}\\ Ae_1 \cdot Ae_1 &= \binom{a_{11}^2}{a_{21}^2}\\ A(e_2 \cdot e_2) &= \binom{a_{12}}{a_{22}}\\ Ae_2 \cdot Ae_2 &= \binom{a_{12}^2}{a_{22}^2}\\ \end{align*} That is, as $x^2 = x$ has the only solutions $x = 0,1$, all entries of $A$ must be either $0$ or $1$. Moreover \begin{align*} A(e_1\cdot e_2) &= \binom{0}{0}\\ Ae_1 \cdot Ae_2 &= \binom{a_{11}a_{12}}{a_{21}a_{22}}\\ \end{align*} That is each row of $A$ must contain at least one zero. This gives us as the only possible matrices $$ \def\m#1#2#3#4{\begin{pmatrix} #1 & #2 \\ #3 & #4\end{pmatrix}} \m 0000, \m 1000, \m 0100, \m 0010, \m 0001, \m 1001, \m 1010, \m 0101, \m 0110 $$ giving as the seven homomorphisms you gave and aditionally the two (the seventh and the eighth matrix above) $$ f(a,b) = (a,a), \qquad f(a,b) = (b,b) $$ Note that if you want your homomorphism to be unitary, that is $f(1,1)= (1,1)$, you have the additional constraint $$ \binom{a_{11} + a_{12}}{a_{21} + a_{22}}=A(e_1 + e_2) = e_1 + e_2 =\binom 11 $$ that is each row must contain exactly one One, giving us the four homomorphisms $$ \m 1001, \m 1010, \m 0101, \m 0110 $$