Given a module M, let P be a projective module with a surjection p \colon P \to M. Then set \Omega(M) to be the kernel of p. Suppose we are given a morphism f \colon M \to N and a surjection q \colon Q \to N where Q is projective.
For certain rings, such as Frobenius algebras, \Omega is an equivalence of categories.