= Semisimplicity from the regular module
If the <left regular module> of a unital <ring> is semisimple, then every left module is semisimple. Every module is a quotient of a direct sum of copies of the regular module, hence a quotient of a sum of <simple modules>. Their simple or zero images span the quotient. A maximal independent direct sum of simple submodules must span: otherwise one of these simple images has zero intersection with it and can be added. Thus the quotient is a direct sum of simples. For an infinite sum the maximality argument uses <Zorn's lemma>.
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