Is prime number defined to be some natural number or integer

Question

In number theory, is prime number usually defined to be some natural number or some integer, i.e., must it be positive or can it be either positive or negative?

Thanks and regards!

Answer 1

Today, the definition depends on context. But the definition that applies to arbitrary rings is usually along the lines of:

A nonzero element $p$ of a ring $R$ is a prime if and only if $p$ is not a unit, and whenever $p|ab$, either $p|a$ or $p|b$.

(Sometimes one also allows $0$, which is a prime number in $R$ if and only if $R$ has no "zero divisors". In the integers, allowing it has virtues and minuses, usually reflected in exclusion clauses in theorems depending on whether you admit it or not).

If we take this definition and apply it to $\mathbb{Z}$, then negative numbers such as $-2$, $-7$, etc., are "primes." This leads to some difficulties with the Fundamental Theorem of Arithmetic (and its generalization to UFDs), in that factorization into primes is no longer unique: $4 = 2\times 2 = (-2)\times(-2)$.

But divisibility also defines an equivalence relation, "are associates": we say that $a$ and $b$ are associates if and only if $a|b$ and $b|a$, which for rings like $\mathbb{Z}$ that are domains (no zero divisors, and a multiplicative identity) is equivalent to saying that there is a unit $u$ such that $a=ub$. So we associate prime elements with each other if they differ by a unit, so that we consider both $2$ and $-2$ to be "essentially the same" for divisibility purposes. The fundamental theorem of arithmetic then says that the factorization is unique "up to order and associates" (so you may replace some primes by other primes that are associates of them, such as replacing $2$ by $-2$).

These days in number theory.... if you are leading towards algebraic number theory, the definition of prime will likely encompass both the positive and negative versions, because it facilitates the passage to things like primes in $\mathbb{Z}[i]$, and the consideration of prime ideals later on. But in very elementary, beginning, number theory when you are dealing only with natural numbers, you'll likely stick to the natural number definition.

Because in any ring $a|b$ if and only if for any units $u$ and $v$ you have $ua|vb$, and because the property of being a prime number has to do with divisibility, it usually doesn't matter whether you are restrictive or not (provided that, if you allow negative numbers as primes, you add an 'up to associates' clause whenever needed).

Answer 2

These 3 answers pretty much cover everything. I agree with all of them, but mostly with #3.