Question

Let R be a commutative ring with identity, and let N be the set of non-units in R. Give an example to show that N need not be an ideal.

Short answer

It is proved that N is not an ideal in R.

Step-by-step solution

Given statement

It is given that R is a commutative ring with identity and Nis the set of all non-units in R.

Proof part

Consider that $R=\mathbf{\mathbb{Z}}$. Whose units are $\pm 1$. So, $N=\mathbf{\mathbb{Z}}-\left\{\pm 1\right\}$.

Notice that, then

$3+\left(-2\right)=1\notin N$

This implies that N is not closed under addition.

Hence, N is not ideal in R.