# Is there a name for rings of the form $R/\mathfrak{q}$ with $\mathfrak{q}$ primary?

In a unital commutative ring $R$, a lot of the times we characterize the property of an ideal $\mathfrak{a}$ by the property of the quotient ring $R/\mathfrak{a}$. Examples are if $\mathfrak{a}$ is prime, maximal, radical then $R/\mathfrak{a}$ is respectively integral domain, field, reduced. In all of these examples we do the following

The ideal $\mathfrak{a}\subset R$ has property P if and only if the zero ideal of $R/\mathfrak{a}$ has property P.

Out of curiosity I asked myself if $\mathfrak{q}$ is primary, then what property does $R/\mathfrak{q}$ have? The answer is:

An ideal $\mathfrak{q}\subset R$ is primary if and only if every zero-divisor of $R/\mathfrak{q}$ is nilpotent. As a result the nilradical of $R/\mathfrak{q}$ is prime under either condition; it is the unique minimal prime ideal of $R/\mathfrak{q}$.

I am wondering whether there are names for these rings:

• A ring $R$ in which all zero-divisors are nilpotent.
• A ring $R$ which has a unique minimal prime ideal.

These rings seem to be rather important, so I would be surprised if there is not a name for them. However googling didn't get me anywhere.

I say "unsurprising" because in general ring theory you say that $R/P$ is a prime ring if $P$ is prime, and $R/S$ is a semiprime ring if $S$ is a semiprime ideal. These boil down to domains and reduced rings for commutative rings, of course.
• Ring theorists tend to give different names to the property on $I$ and on $R/I$. $$\begin{array}{|r|r|}\hline I & R/I\\\hline\text{maximal}&\text{field}\\\hline\text{prime} & \text{integral domain} \\\hline\text{radical}&\text{reduced}\\\hline\text{primary} & ? \\\hline \end{array}$$ I think we could do better and look a different name for the missing slot. My humble proposal is to call integrary domain to a ring in which every zerodivisor is nilpotent. Commented May 22 at 12:39
• @ElíasGuisadoVillalgordo It depends on how you pick your adjectives. A noncommutative theorist would write $\begin{array}{|r|r|}\hline I & R/I\\\hline\text{prime ideal} & \text{prime ring} \\\hline\text{semiprime ideal}&\text{semiprime ring}\\\hline\text{primary ideal} & \text{primary ring} \\\hline \end{array}$. The only hitch is that "primary ideal" is not quite as standard in noncommutative context. Commented May 22 at 13:51