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

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In summary, we discussed the two concepts of ring semisimplicity, left and right, for non-unital rings. While they coincide for commutative rings, there may be different results for non-commutative rings that do not have a unit element. There may also be structure theorems for non-unital semisimple rings, although further research is needed to determine the specific results.
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Ruslan_Sharipov
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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?

Are there any structure theorems for non-unital semisimple rings?
 
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Ruslan_Sharipov said:
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##.
 

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

What is a semisimple ring?

A semisimple ring is a type of ring in abstract algebra that is characterized by having no nontrivial two-sided ideals. This means that every nonzero element in the ring has a multiplicative inverse, and the ring can be decomposed into a direct sum of simple rings.

What is the significance of semisimple rings in mathematics?

Semisimple rings are important in the study of representation theory and algebraic geometry. They also have applications in areas such as group theory and quantum mechanics.

How do semisimple rings differ from simple rings?

While both semisimple and simple rings have no nontrivial two-sided ideals, semisimple rings are further characterized by being a direct sum of simple rings. Simple rings, on the other hand, cannot be decomposed in this way.

Can semisimple rings have zero divisors?

No, semisimple rings cannot have zero divisors. This is because every nonzero element has a multiplicative inverse, so there is no element that can multiply with another element to equal zero.

What is the relationship between semisimple rings and semisimple modules?

A semisimple ring is closely related to semisimple modules, as every semisimple module over a ring is also a semisimple ring. This means that the properties and structure of semisimple rings can also be applied to semisimple modules.

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