TITLE: Primes in ring theory
QUESTION [5 upvotes]: I have this definition of primes in my lecture notes and I don't understand it i was wondering whether someone could explain it to me
$p$ is prime if and only if for all $a$, $b$ that are elements of $R$, we have $p \mid ab$ implies $p \mid a$ or $p \mid b$.
See I am rather lost by this definition.

REPLY [4 votes]: Have you tried seeing why composite numbers don't fit that definition?
For example, take $p = 14$, $a = 2$, $b = 49$. Then $ab = 98$ is divisible by $p$. However, $2$ is clearly not divisible by $14$, nor is $49$ divisible by $14$ either. No surprise, given that $14$ is composite, not prime.
Now consider $p = -7$, $a = 2$, $b = 49$. Then $ab = 98$ is divisible by $p$, just like before. Although $2$ is not divisible by $-7$, $49$ is. We can choose other $a$ and $b$ such that $ab = 98$, but it is guaranteed that in each case we will find that $-7$ is a divisor of either $a$ or $b$, if not both. Indeed $-7$ is a prime number.
Working in $\mathbb{Z}$ this might not seem like such a big deal. Look at almost any other number ring and this becomes a very important distinction.