I am interested in semisimple rings and semisimple modules which are not unital. There are two concepts of ring semisimplicity: left semisimplicity and right semisimplicity. A ring is called semisimple on the left if it is presented as a sum of its simple left ideals. A ring is called semisimple on the right if it is presented as a sum of its simple right ideals. Do these two concepts coincide in the case of non-unital rings? Are there any structure theorems for non-unital semisimple rings?