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 $X_2$?
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.
Thanks to the @bungo, I add more conditions