Question

Is the subset $\left\{1,-1,i,-i\right\}$a subring of $\mathrm{ℂ}$.

Short answer

Hence, $\left\{1,-1,i,-i\right\}$ is not a subring of $\mathrm{ℂ}$.

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$\mathrm{ℝ}$ .

Proving Axioms

The given subset is:

$\left\{1,-1,i,-i\right\}$

Clearly,$i+i=2i\notin \left\{1,-1,i,-i\right\}$ as an axiom for addition is not satisfied here.

Also, $0\notin \left\{1,-1,i,-i\right\}$.

Hence, $\left\{1,-1,i,-i\right\}$ is not a subring of $\mathrm{ℂ}$.