Universal central extensions of precrossed modules and Milnorʹs relative K2

We characterize the universal central extension of a perfect precrossed module giving two descriptions, one in terms of non-abelian tensor products of groups and other in terms of projective presentations. As application to relative algebraic K-theory, we obtain that Milnorʹs absolute and relative K2 groups are the kernel of the universal central extension of the precrossed module determined by the groups of the elementary matrices of a ring and relative to an ideal, respectively.