Prime elements

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

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