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In the book 'Spinors, Clifford and Cayley Algebras' Hermann states that any (finite dimensional) semisimple associative algebra is the direct sum of minimal left ideals. Here, 'semisimple' is defined as 'has no nilpotent ideals except $0$'. However, in the proof he seems to use an unit, which was not in the assumptions.

Interestingly enough, later in the book it is mentioned as an exercise that semisimple algebras have an unit, but I think that exercise uses the earlier theorems.

Therefore, my questions are:

  • Is Hermann's statement generally true without assuming the algebra is unital?
  • If it is true, how can it be proven?

Thanks in advance!

Pepijn de Maat
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  • Well, this much is true, there is a way to adjoin unity to a ring $R$ such that the resulting larger unital ring $R'$ has an isomorphic copy of $R$ embedded. For example, https://math.stackexchange.com/questions/1175193/adjoining-1-to-a-ring So, if his proof survives this construction you might be able to transport back the resulting decomposition into the direct product of simple rings. – James S. Cook Mar 24 '18 at 14:12
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    The theorem seems to basically say that “semisimple according to Hermann's definition” is the same as “semisimple according to the definition everybody else uses“. – egreg Mar 24 '18 at 15:13

2 Answers2

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This isn’t even true for rings with identity, as stated. Any local domain that isn’t a division ring is “semisimple” by that definition, but since it’s a domain that isn’t and vision ring, it doesn’t have any minimal ideals at all.

Perhaps the author additionally assumes the artinian condition? Or finite dimensionality?

rschwieb
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After re-reading it, I think that the `unit' was just a strange abuse of notation. The unit never existed by itself, but only in products such as $A(1-e)$. By replacing all instances of terms like $A(1-e)$ by $\{a - ae \mid a \in A\}$, the proof is 'fixed'.

The (finite dimensional) semisimple associative algebra $A$ indeed has a unit, however, to proof that we need that $A$ is the direct sum of left ideals which are generated by minimal idempotents, or we need an argument where we adjoin a unit to $A$ (which I haven't checked, but probably also relies on the sum of left ideals).

Pepijn de Maat
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