I am confused in the deduction from the text, "Algebraic Number Theory", by Taylor. On p51, the author claimed 1,2, below as a direct consequence of 0.
0. Let $A$ denote a finite dimensional commutative $F$-algebra. Then $Rad(A)=0$ iff $A \cong \prod A_i$ with each $A_i$ a field.
1. If $A$ is a finite dimensional commutative $F$- algebra, then it has finitely many maximal ideals. 2. If $A$ is also an integral domain, then it is a field.
I could prove 1,2 individually from artinian proeprties. But I am quite confused why $0 \Rightarrow 1,2$.
As written in the comments,take $B=A/Rad(A)$ and use $0$. This tells you that $B$ as a finitely generated algebra is s field product,then has only finitely many maximal ideals. Also maximal ideals of $B$ are exactly the ones of $A$. If $A$ is a domain, $Rad(A)=0$ and then use $0$ again and that a product of rings is not likely to be a domain.