Question

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

Short answer

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

Step-by-step solution

Rings

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

Finding subring

The given set is:

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

Let the elements be:

$\left(-1,1\right)$and$\left(-2,2\right)$

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

$\left(-1,1\right)\left(-2,2\right)=\left(2,2\right)\ne 0$

The closure property is not satisfied.

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