Left ideals vs right ideals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:08:03Z http://mathoverflow.net/feeds/question/92852 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92852/left-ideals-vs-right-ideals Left ideals vs right ideals Jan Veselý 2012-04-01T22:19:16Z 2012-04-01T23:09:05Z <p>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).</p> <p>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?</p> http://mathoverflow.net/questions/92852/left-ideals-vs-right-ideals/92857#92857 Answer by miguel for Left ideals vs right ideals miguel 2012-04-01T23:09:05Z 2012-04-01T23:09:05Z <p>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:</p> <p>Tsit-Yuen Lam, A first course in noncommutative rings.</p>