Apologies in advance if this was asked before.
Say we have $X$ is embedded into $Y$, my understanding is that there exists an injective ring homomorphism from $X$ to $Y$. I have the following questions about the idea of 'embedding': (as I want to get more familiar with this terminology)
$1)$ Is the idea of embedding, loosely speaking, saying that there is a subring in $Y$ that is isomorphic to $X$?
$2)$ I saw this question earlier today, where the answer says 'properly, you get an embedding $\mathbb N\to K$'. However though, can the naturals be embedded to an arbitrary field $K$ if $K$ is finite?
$3)$ Lastly, maybe just a check of my understanding, is an inclusion map always an embedding map?