Normal subgroup and ideal inclusion

by se-hyuck yang   Last Updated July 16, 2019 09:20 AM - source

I'm a student who just started abstract algebra.

There is an easy question for you.

Let $$X_1 \le Y$$ which means $$X_1$$ is a subobject(subring or subgroup) of $$Y$$

Also let $$X_2$$ bes a (normal or ideal) of $$Y$$

According to the (group or ring) $$2nd$$ isomorphism theorem, We can easily pull out the conclusion that $$X_1 \cap X_2$$ is an ideal of or normal in $$X_1$$.

But Question is

If the sets $$X_1$$ and $$X_2$$ are not contained in each other

(I.E. There aren't case that $$X_1 \subset X_2$$ or $$X_2 \subset X_1$$)

And not disjoint from each other then...

$$(1)$$ Considering group case, is $$X_1 \cap X_2$$ a normal subgroup of $$X_2$$?

Considering ring case, is $$X_1 \cap X_2$$ an ideal of $$X_2$$?

$$(2)$$

Considering group case, ie $$X_1 \cap X_2$$ is a normal subgroup of $$Y$$?

Considering ring case, Does $$X_1 \cap X_2$$ is a ideal of the $$Y$$?

Whenever I try proving n (1) and (2), they look like true.

But I don't have confidence that both of them bre true.

Any help would be appreciated.

Thank you.

Thanks to the @bungo, I add more conditions

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