An element $p$ of a commutative ring $R$ is called "prime" if, for any $a,b\in R$, whenever $ab$ is a multiple of $p$, either $a$ or $b$ is a multiple of $p$.

Is there a word for the "prime-like" property that, whenever $ab$ is a multiple of $p^2$, either $a$ or $b$ is divisible by $p$? Or another, more usual concept in ring theory that this is connected to?

I ask because the "prime-likeness" of $2$ in $R$ seems to control whether the quadratic formula can be made to work for monic polynomials over $R$ (as long as $2$ is also not a zero-divisor). This is because, if the discriminant of $x^2 + bx + c$ is a square $b^2 - 4c = d^2$, then $(-b+d)(-b-d) = 4c$, so at least one (and hence both) of $(-b+d)$ and $(-b-d)$ are multiples of $2$ in $R$. Their halves are the two roots of $x^2 + bx + c$.

For example, $2$ is "prime-like" in $\mathbb{Z}[\sqrt{2}]$, which is easy to verify elementarily. Hence a monic quadratic over $\mathbb{Z}[\sqrt{2}]$ factors iff its discriminant is a square. But $2$ is not "prime-like" in $\mathbb{Z}[\sqrt{5}]$, since $(\sqrt{5}-1)(\sqrt{5}+1) = 4$. And indeed, the discriminant of $x^2 -x-1$ is a square in $\mathbb{Z}[\sqrt{5}]$, but the polynomial doesn't factor there.