Exercise 6, § 1

Let $A$ be a commutative algebra over a commutative field and let $I$ be a maximal ideal of $A$. Prove that one has $A^2\subset I$ with dim(A$/I$) $= 1$, or else that $A/I$ is a field. (Apply to $A/I$ Exercise 3 of A, I, p. 153.) In particular, if $A^2= A$, every maximal ideal of A is regular.