Semisimple Frobenius Algebras

Question

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

Answer 1

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.