Goldie's theorem for semigroups

In this paper, we try to look for a semigroup analogue of Goldie's theorem which characterizes the orders of the full matrix ringM _n (D) over a division ringD. The main result of this paper, Theorem 4.11, does give a characterization of the semigroups which have a classical left quotient semigroupM _n (G ⁰), whereG ⁰ is a group with zero. The author is grateful to the referee for his careful modification of this paper and helpful comments which resulted in several improvements.