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

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.

• And $N$ is a radical ideal if and only if $R/N$ is a reduced ring, etc. Commented Aug 4 at 18:38
• That seems to be a pretty good characterisation. I'm not sure if you can say anything better than that "iff."; $Q$ is primary iff. $R/Q$ has a particular structure Commented Aug 4 at 18:48

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.