# Ideals - Intersection of Ideals

• Mar 30th 2013, 05:40 PM
Bernhard
Ideals - Intersection of Ideals
If I and J are ideals of R, prove that their intersection $I \cap J$ is also an ideal of R

Hence show that the intersection of an arbitrary nonempty is again an ideal (do not assume that the collection is countable)

Would appreciate help with this exercise ... my main problem is with generalising to an uncountable collection

Peter
• Mar 31st 2013, 02:54 AM
Gusbob
Re: Ideals - Intersection of Ideals
Quote:

Originally Posted by Bernhard
If I and J are ideals of R, prove that their intersection $I \cap J$ is also an ideal of R

Hence show that the intersection of an arbitrary nonempty is again an ideal (do not assume that the collection is countable)

Would appreciate help with this exercise ... my main problem is with generalising to an uncountable collection

Peter

There should be nothing different in the uncountable case save for notation (which is possible to hide in a proof likes this). How did you prove the other cases?
• Mar 31st 2013, 05:45 PM
Bernhard
Re: Ideals - Intersection of Ideals
I and J ideals of R $\longrightarrow I \cap J$ is an ideal of R

Proof

$x, y \in I \cap J$

$\longrightarrow$ $x, y \in I , x,y \in J$

$\longrightarrow$ $x - y \in I , x - y \in J$

$\longrightarrow$ $x - y \in I \cap J$ .... (1)

$x \in I \cap J$

$\longrightarrow$ $x \in I , x \in J$

$\longrightarrow$ $rx \in I , rx \in J$ for all $r \in R$

$\longrightarrow$ $rx \in I \cap J$ ..... (1)

(1) , (2) $\longrightarrow$ $I \cap J$ is ideal of R

Then since $I \cap J$ is an ideal, use the above proof on K and $I \cap J$ to show $I \cap J \cap K$ is an ideal and so on for countably many ideals.
• Mar 31st 2013, 07:48 PM
Gusbob
Re: Ideals - Intersection of Ideals
Given an uncountable indexing $\Lambda$ of ideals, I modified your proof for the uncountable case. If you take $\Lambda$ to be a countable or finite indexing, you only really need this one proof.

Quote:

Originally Posted by Bernhard
Proof

$x, y \in \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha}$

$\longrightarrow$ $x, y \in I_\alpha$ for all $\alpha \in \Lambda$

$\longrightarrow$ $x - y \in I_\alpha$ for all $\alpha \in \Lambda$

$\longrightarrow$ $x - y \in \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha}$ .... (1)

$x \in I \cap J$

$\longrightarrow$ $x \in I_\alpha$ for all $\alpha \in \Lambda$

$\longrightarrow$ $rx \in I_\alpha$ for all $\alpha \in \Lambda$ and for all $r \in R$

$\longrightarrow$ $rx \in \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha}$ ..... (1)

(1) , (2) $\longrightarrow$ $\displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha}$ is ideal of R