Prime elements

This article page is a stub, please help by expanding it.

A prime element of a domain is an element $p$ which is neither zero nor a unit divisible only by units and associates and which also satisfies the following condition: if $p|ab$, where $a$ and $b$ are also in the same domain, then either $p|a$ or $p|b$, maybe both; but if neither of those holds true, then $p$ may be irreducible but it is not prime. In fact, if the domain is not a unique factorization domain, it does not have prime elements though it may have irreducible elements.

For example, in $\mathbb{\mathbb{Z}}[i]$ (see: Gaussian integers), we see that $(1-i)|(2\times 5)$ and that $\frac{2}{1-i}=1+i$. Though $\frac{5}{1-i}=\frac{5+5i}{2}\in ̸\mathbb{\mathbb{Z}}[i]$, this does not detract from the fact that $1-i$ is a prime element of $\mathbb{\mathbb{Z}}[i]$.

Although 2 is irreducible in $\mathbb{\mathbb{Z}}[\sqrt{-5}]$, it is not prime. For example, $2|((1-\sqrt{-5})(1+\sqrt{-5}))$, but $\frac{1-\sqrt{-5}}{2}=\frac{1}{2}-\frac{\sqrt{-5}}{2}\in ̸\mathbb{\mathbb{Z}}[\sqrt{-5}]$ and $\frac{1+\sqrt{-5}}{2}=\frac{1}{2}+\frac{\sqrt{-5}}{2}\in ̸\mathbb{\mathbb{Z}}[\sqrt{-5}]$ either. Note that 6 has two factorizations into irreducibles: $6=2\times 3=(1-\sqrt{-5})(1+\sqrt{-5})$.