Is the difference of ideals an ideal?

Question

I am studying ideals and noticed that $I+J$ is an ideal as noted here. However the paper does not discuss $I-J$ so:

Is the difference of ideals an ideal?

Answer 1

Yes, of course, because

$$I-J = \{i - j \mid i \in I, j \in J\} = \{i + (-j) \mid i \in I, j \in J\} = \{i + k \mid i \in I, -k \in J\} = \{i + k \mid i \in I, k \in -J\} = \{i + k \mid i \in I, k \in J\} = I + J$$

(because obviously $-J = J$).