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I'm reading this paper and am stuck on the proof of theorem 3.4 on page 7. In particular, how does the author justify the statement below

If some component A′ of A is not a field, then it contains nontrivial nilpotents

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Context: $A$ is assumed to be a commutative Frobenius algebra over some field $k$, and $A^{\prime}$ is a factor in the decomposition of $A$ as a product of local $k$-algebras.

The author's claim then follows since (as a finite-dimensional $k$-algebra) $A^{\prime}$ as Artinian local, hence its maximal ideal is nilpotent. Therefore $A^{\prime}$ is a field if and only if its maximal ideal is the zero idealif and only if it has no nontrivial nilpotents.

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