If $A$ and $B$ are ideals of a ring $R$. Then $A+B$ is an ideal of $R$ generated by $A \cup B$?

Question

I have proved that $A+B$ is an ideal of $R$. But I'm not able to prove that it is generated by $A \cup B$.

Answer 1

Guide:

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)