  Let $A$ be a not necessarily commutative monoid with zero such that
projective $A$-acts are free. This paper shows that the algebraic K-groups of
$A$ can be defined using the +-construction and the Q-construction. It is shown
that these two constructions give the same K-groups. As an immediate
application, the homotopy invariance of algebraic K-theory of certain affine
$\mathbb{F}_1$-schemes is obtained. From the computation of $K_2(A),$ where $A$
is the monoid associated to a finitely generated abelian group, the universal
central extension of certain groups are constructed.
