Do Left and Right Semisimplicity Coincide in Non-Unital Rings?

[Ruslan_Sharipov]

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?

 

[fresh_42]

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?

They coincide for commutative rings. $1$ has nothing to do with it here.

Are there any structure theorems for non-unital semisimple rings?

Very likely, but I don't know much about those rings. Also the standard results for semisimple rings might not - at least not all - use $1\in R$.