= Semisimplicity of a classical quotient from regular elements in essential ideals
Suppose $R$ 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> $Q$, every <right ideal> is generated by its contraction to $R$, making $Q$ right <Noetherian>. An <essential right ideal> of $Q$ contracts to an essential right ideal of $R$, hence contains a regular element and is all of $Q$. 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 $Q$ semisimple Artinian.
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