I have proved that $A+B$ is an ideal of $R$. But I'm not able to prove that it is generated by $A \cup B$.
closed as off-topic by Javi, YuiTo Cheng, Arnaud D., Paul Frost, callculus May 6 at 17:42
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Ideal $J$ is by definition the ideal generated by $A\cup B$ iff
- $J$ is an ideal.
- $A\cup B\subseteq J$.
- If $I$ is an ideal with $A\cup B\subseteq I$ then $J\subseteq I$.
Now try to prove $J=A+B$ has these properties in the special case where $A,B$ are ideals .
(You said that you already proved yourself that it has the first property)