Question

Find the subring of the ring Zx Zthat is not an ideal of Zx Z

Answer 1

Question 1.

Find the subring of the ring $\mathbb{Z}\times\mathbb{Z}$ that is not an ideal of $\mathbb{Z}\times\mathbb{Z}$.

Solution. Consider the subset

$D=\{(n,n)\mid n\in\mathbb{Z}\}\subset\mathbb{Z}\times\mathbb{Z}.$

It is a subring of $\mathbb{Z}\times\mathbb{Z}$. Indeed, it is a subgroup of additive group of $\mathbb{Z}\times\mathbb{Z}$, because

$(n,n)-(m,m)=(n-m,n-m)\in D$

for arbitrary $(n,n),(m,m)\in I$. Furthermore, $D$ is closed under multiplication:

$(n,n)\cdot(m,m)=(nm,nm)\in D$

for all $(n,n),(m,m)\in I$. Finally, $D$ contains the identity $(1,1)$ of $\mathbb{Z}\times\mathbb{Z}$. The last fact also shows that $D$ cannot be an ideal of $\mathbb{Z}\times\mathbb{Z}$ because $D\neq\mathbb{Z}\times\mathbb{Z}$. This can be demonstrated by the following observation:

$(1,1)\cdot(1,0)=(1,0)\not\in D,$

while $(1,1)\in D$.

Answer: for example, $D=\{(n,n)\mid n\in\mathbb{Z}\}\subset\mathbb{Z}\times\mathbb{Z}$. $\Box$