What is the structure of quotient ring $R/N$ where $N$ is primary ideal?

Question

Let $R$ be a commutative ring with $1$. Everyone knows the following statement:

1. $R/N $ is field $\iff$ $N$ is maximal ideal.
2. $R/N$ is integral domain $\iff$ $N$ is prime ideal.

I am wondering that what happens if $N$ is primary ideal? What does the $R/N$ structure look like where $N$ is primary ideal of $R$?

What is the quotient ring $R/N$ where $N$ is primary ideal?

I only know the following theorem about ring quotient by primary ideal.

Let $R$ be a commutative ring with identity. A proper ideal $Q$ of $R$ is primary if and only if each zero-divisor in $R/Q$ is nilpotent.

Answer 1

Traditionally, if one has some condition one calls "foo ideal," it's customary to also say a "foo ring" is something of the form $R/I$ where $I$ is a foo ideal.

ideal $I$	ring $R/I$	also known as (commutative rings)
prime	prime	integral domain
semiprime	semiprime	reduced
right semiprimitive	right semiprimitive	field
maximal	simple	field
Goldman		Goldman domain
primary		primary ring

As you can see, the convention isn't always strictly followed.

Really the "theorem" you mentioned is just a restatement of the defintion of the zero ideal being semiprimary. The same is true for the notions of "prime/semiprime/primary" above.