Q: What is the irreducible head of a principal indecomposable $R$-module $M$? What is the irreducible head of a principal indecomposable $R$-module $M$?
and some properties of the irreducible head of $R$-module $M$?
Thanks to everyone!

A: The head (or top) of a module is defined to be $M/J(M)$ where $J(M)$ denotes the Jacobson radical of the module.
When $R$ is a perfect ring, then the head/top of every principal indecomposable module is a simple module (which is what I think you mean by irreducible module). Not sure what other properties you are interested in. Obviously it can't be projective unless $J(M)=\{0\}$, since $M$ is a projective cover for its head.
Did you read the wiki that outlines all of this?
