A left Artinian ring that is also a right Noetherian ring

[frankusho]

1. Prove that a ring which is left Artinian and right Noetherian is right Artinian.

The Attempt at a Solution

I can't figure it out. Can anyone help?

 

[Mark44]

What are the definitions of these terms?

 

[frankusho]

A left Artinian ring R is a ring for which every descending chain R=I₀ ⊃I₁ ⊃I₂ ⊃…⊃I_n ⊃… of its left ideals stabilizes, i.e. there is a k such that I_n+1 =I_n for all n≥k

A right Noetherian ring R is ring in which every ascending chain of right ideals stabalizes

 

[Mark44]

So if a ring is left Artinian and right Noetherian, what can you say? What would you like to have happen to conclude that this ring is also right Artinian?

 

[Mark44]

When you post a question here, include information like what you have below in your initial post. That's why the 2nd section on relevant equations and definitions is there in the template. It should just be erased.

A left Artinian ring R is a ring for which every descending chain R=I₀ ⊃I₁ ⊃I₂ ⊃…⊃I_n ⊃… of its left ideals stabilizes, i.e. there is a k such that I_n+1 =I_n for all n≥k

A right Noetherian ring R is ring in which every ascending chain of right ideals stabalizes