Question

Is $\mathit{S}\mathbf{=}\mathbf{\left\{}\mathbf{(}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{)}\mathbf{|}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{=}\mathbf{0}\mathbf{\right\}}$a subring of$\mathbb{Z}\times \mathbb{Z}$ ? Justify your answer.

Short answer

The given set is not a subring of $\mathbb{Z}\times \mathbb{Z}$.

Step-by-step solution

Rings: 

A ringcan be defined as a non-empty set $ℝ$whose elements deal with basically two operations:additionand multiplication, where elements belong to the real number $\mathrm{ℝ}$.

Finding subring:

The given set is:

$S=\left\{(a,b)|a+b=0\right\}$

Let the elements be:

$(-1,1)\text{and}(-2,2)$.

Now, if $S$ is a ring, then:

.$(-\text{1},\text{1})(-\text{2},\text{2})=(\text{2},\text{2})\ne \text{0}$

The closure property is not satisfied.

Hence, the given set is not a subring of$\mathbb{Z}\times \mathbb{Z}$.