Why in a graded ring $A$ finitely generated that's an algebra over a field $K$ every maximal ideal is a $K$-subspace?

Question

Probably this question has already been asked, but I'm very bad in find old question and I searched for half an hour, so I'm asking it again.

I suppose that's true beacuse my professor used this fact.

Answer 1

For any algebra $A$ over $K$, an $A$-module $M$ is a $K$-vector space by restriction of scalars, and an ideal of $A$ is an $A$-submodule of $A$. This has nothing to do with grading of $A$ nor finitely generated nor maximal ideal.

Answer 2

The fact is more general: Every algebra is a $K$-vector space and every subalgebra is a $K$-subspace. In particular, every ideal.