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


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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)

  • $\begingroup$ Thanks for the help! And yes I've proved the first property but now dont know how to prove the other two.🙁 $\endgroup$ – Mariam May 5 at 12:45
  • $\begingroup$ Do you agree that $A\subseteq A+B$ and $B\subseteq A+B$? That gives the second property. $\endgroup$ – drhab May 5 at 12:47
  • $\begingroup$ Yes!! Okay and what about the third one? $\endgroup$ – Mariam May 5 at 12:51
  • $\begingroup$ It remains to prove that $A,B\subseteq I$ implies that $A+B\subseteq I$ for any ideal $I$. Well, elements of $A+B$ have the shape $a+b$ with $a\in A\subseteq I$ and $b\in B\subseteq I$. Then it follows that $a+b\in I$ since ideal $I$ is closed under addition. $\endgroup$ – drhab May 5 at 12:55
  • $\begingroup$ It's okay.I got it.Thanks a lot for clearing my doubts! $\endgroup$ – Mariam May 5 at 13:08

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