Morita equivalence and Morita duality for rings with local units and subcategory of projective unitary modules

We study Morita equivalence and Morita duality for rings with local units. We extend the Auslanders results on the theory of Morita equivalence and the Azumaya-Morita duality theorem to rings with local units. As a consequence, we give a version of Morita theorem and Azumaya-Morita duality theorem over rings with local units in terms of their full subcategory of finitely generated projective unitary modules and full subcategory of finitely generated injective unitary modules.