PID PID 2001-11-04 22:55:34 2007-05-27 19:22:19 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A \emph{principal ideal domain} is an integral domain where every ideal is a principal ideal. In a PID, an ideal $(p)$ is maximal if and only if $p$ is irreducible (and prime since \PMlinkname{any PID is also a UFD}{PIDsAreUFDs}). Note that subrings of PIDs are not necessarily PIDs. (There is an example of this within the entry biquadratic field.)