Projective cover ($P(M)$)

A projective cover of a module $M$ is a surjective module homomorphism $p:P\to M$ with $P$ a projective module and with superfluous kernel: if a submodule $L\subseteq P$ satisfies $L+\ker p=P$, then $L=P$. Equivalently, no proper submodule of $P$ still maps onto $M$. This is the meaning of an essential surjection in this definition.

For a finite-dimensional algebra, every finite-dimensional module has a projective cover. The cover of a simple module is a principal indecomposable module and has that simple module as its head of a module. For example, over the dual numbers $k[\epsilon ]/({\epsilon }^2)$, the quotient map to $k$ is a projective cover: any element lifting a nonzero residue is a unit and generates the whole algebra. A general projective surjection need not have the minimality property.

The head, or top, of a finite-length module $M$ is its largest semisimple quotient. It is $M/J(M)$, where $J(M)$ is the radical of a module.