Suppose is a right Noetherian ring and every essential right ideal contains a regular element of a ring. Preimages of essential regular principal right ideals under left multiplication are essential, so they contain regular elements and prove the right Ore condition. In the classical right ring of quotients , every right ideal is generated by its contraction to , making right Noetherian. An essential right ideal of contracts to an essential right ideal of , hence contains a regular element and is all of . A maximal disjoint complement to any right ideal has essential sum with it, so every right ideal is complemented. The right regular module is therefore a finite direct sum of simple modules, making semisimple Artinian.

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  1. Goldie's theorem
  2. Right Goldie ring
  3. Goldie ring
  4. Noncommutative algebra
  5. Algebra
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