I'm studying abstract algebra, with Dummit's book.
Our professor introduced a lemma, and I'm confused with some concept.
Here is the lemma: "Let $R$ be an integral domain, and let $p$ be in $R$. Then $p$ is prime if and only if $R_p$ (principal ideal generated by $p$) is a nonzero prime ideal."
The proof of this lemma says that $p$ is prime if and only if $ab$ is divisible by $p$ then $x$ or $y$ is also divisible by $p$, and it is same with $ab$ in $R_p$ implies that $a$ or $b$ is also in $R_p$, and it is same as $R_p$ is a prime ideal.
I don't know why "$p$ is prime if and only if $ab$ is divisible by $p$ then $x$ or $y$ is also divisible by $p$, and it is same with $ab$ in $R_p$ implies $a$ or $b$ also in $R_p$." What is the principal ideal generated by $p$? Its mean is every ideal of $R$ is generated by $p$, right? But how we can say that $ab$ in $R_p$ implies that $a$ or $b$ is also in $R_p$?
For any given element $x$ of a commutative unital ring $R$, the ideal generated by $x$ is just the set $Rx$ of its multiples. E.g. the principal ideal of $2$ in $\Bbb Z$ consists of the even numbers.
Now, $ab\in Rp$ is the same as saying $p$ divides $ab$, and thus the condition that $Rp$ is a prime ideal coincides with the condition that $p$ is a prime element.
Note also that ideals generalize the concept of 'being the multiple of something(s)'.