If is a reduced ring and has only finitely many idempotents then prove that is a direct product of a finite number of indecomposable zero-dimensional reduced rings i.e. fields .
but then we wouldn't need the condition "R is reduced" anymore because it would be a result of the conditions "R is commutative" and "every principal ideal of R is an idempotent". here's why:
suppose then, since we'll get i.e. R is reduced. anyway, i'll assume the problem is this:
Claim: let R be a commutative ring with 1. if the number of idempotents of R is finite and every principal ideal of R is an idempotent ideal, then R is a direct product of finitely many fields.
Proof of the claim:
1) where is the Jacobson radical of .
Proof: let since we have for some thus and hence because is a unit of R.
2) every principal ideal of R is generated by an idempotent.
Proof: let since we have for some then is an idempotent and it's easy to see that
3) R is artinian.
Proof: since the number of idempotents of R is finite, the number of principal ideals of R must also be finite, by 2). so the number of ideals of R is finite and thus R is artinian.
4) R is a direct product of finitely many fields.
Proof: by 1) and 3), R is a commutative semisimple ring and therefore, by Wedderburn-Artin theorem, R is a direct product of finitely many fields.