A question on semisimple rings

  • #1

Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
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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##.
 

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