Question

Let R be a ring, I an ideal of R, J an ideal of I. Show that if J has a unity, then J is

an ideal of R. Also give an example to show that if J does not satisfy this condition

it need not be an ideal of R.

Answer 1

A)

Let $j \in J$ and $r\in R$.

Also, let e be the unity of J.

Then, $re \in I$ (since I is an ideal of R and e is also in I)

And so $(re)j \in J$ (since J is an ideal of I)

But $(re)j = rj$

i.e $rj \in J$

Hence, J is an ideal of R.

B) Consider $\mathbb{Z_4}$ and $\{0,2\}$

Let R = $\mathbb{Z_4}$ , I $= \mathbb{Z_4}$ and J = $\{0,2\}$

We see from this example that the condition above will not be satisfied because, J has defined here is only a sub ring and not an ideal.