I've just seen a corollary in P.Etingof's book Introducion to Representation Theory which states that every irreducible finite-dimensional representation of a commutative algebra is 1-dimensional:
The similar result I've come across before, characterizing irreducible finite-dimensional representations of abelian groups, requires a representation over an algebraically closed field, while Corollary 2.3.12. seems to apply to any field as shown in the screenshot.
Besides, the author uses Schur's lemma(Corollary 2.3.10.), which requires the base field to be algebraically closed, to prove Corollary 2.3.12. as in the following two pictures:
So I wonder if Corollary 2.3.12 is a stronger version of the proposition characterizing irreducible finite-dimensional representations of abelian groups over an algebraically closed field, or that the statement of this corollary is incomplete? Thanks in advance!