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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