# Thread: Union ideals

1. ## Union ideals

Hl!. I have problems with exercice

Let $\displaystyle I$ and $\displaystyle J$ be ideals in a commutative ring $\displaystyle R$ . Prove that their union $\displaystyle I \cup{J}$ is a ideal if and only if $\displaystyle I\subseteq{J}$ or $\displaystyle J\subseteq{I}$

Thanks

2. ## Re: Union ideals

If y is in J but not in I and x is in I but not in J, consider x+y

3. ## Re: Union ideals

Suppose i is in I but not in J. Suppose j is in J but not in I.

If $\displaystyle I\cup J$ is an ideal, then $\displaystyle i+j\in I\cup J$, but $\displaystyle i+j\in I$ implies ________ and $\displaystyle i+j\in J$ implies _________. Any contradictions?

I will admit my knowledge of commutative algebra is less than ideal these days.

Thanks