PID

A principal ideal domain is an integral domain where everyideal is a principal ideal.

In a PID, an ideal $(p)$ is maximal if and only if $p$ is irreducible(and prime since any PID is also a UFD (http://planetmath.org/PIDsAreUFDs)).

Note that subrings of PIDs are not necessarily PIDs. (There isan example of this within the entry biquadratic field.)