Let $R$ be a ring. A two-sided proper ideal $\mathfrak{p}$ of a ring $R$ is called a prime ideal if the following equivalent conditions are met:

1. 1.

   If $I$ and $J$ are left ideals and the product of ideals $IJ$ satisfies $IJ\subset\mathfrak{p}$, then $I\subset\mathfrak{p}$ or $J\subset\mathfrak{p}$.

2. 2.

   If $I$ and $J$ are right ideals with $IJ\subset\mathfrak{p}$, then $I\subset\mathfrak{p}$ or $J\subset\mathfrak{p}$.

3. 3.

   If $I$ and $J$ are two-sided ideals with $IJ\subset\mathfrak{p}$, then $I\subset\mathfrak{p}$ or $J\subset\mathfrak{p}$.

4. 4.

   If $x$ and $y$ are elements of $R$ with $xRy\subset\mathfrak{p}$, then $x\in\mathfrak{p}$ or $y\in\mathfrak{p}$.

$R/\mathfrak{p}$ is a prime ring if and only if $\mathfrak{p}$ is a prime ideal. When $R$ is commutative with identity, a proper ideal $\mathfrak{p}$ of $R$ is prime if and only if for any $a,b\in R$, if $a\cdot b\in\mathfrak{p}$ then either $a\in\mathfrak{p}$ or $b\in\mathfrak{p}$. One also has in this case that $\mathfrak{p}\subset R$ is prime if and only if the quotient ring $R/\mathfrak{p}$ is an integral domain.