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By default, let all algebras be complex and unital. I am concerned with the non-commutative algebras. I am wondering if the following might be true (at least for some classes of algebras, like semi-simple algebras).

Suppose $A$ is a complex, non-commutative algebra with a maximal right ideal which is not finitely generated as a right ideal. Must $A$ contain a maximal left ideal which is not finitely generated as a left ideal? Is there any relation between generation of right/left ideals in general?

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The answer is No. A ring is right (left) noetherian if all of its right (left) ideals are finitely generated. You can find an example of a ring R that is right noetherian but not left noetherian in:

Tsit-Yuen Lam, A first course in noncommutative rings.

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