Final answer:
For a commutative ring R where all finitely generated modules are free, an arbitrary non-zero element would generate a principal ideal which is itself a free R-module. This implies every non-zero element is a unit, making R a division ring, and because R is commutative, it must be a field. If no such non-zero elements exist, R is the zero ring.
Step-by-step explanation:
Proof that a Ring where all Finitely Generated Modules are Free is a Field or the Zero Ring
To prove that if every finitely generated module over a commutative ring R is free, then R must be a field or the zero ring, consider the following argument:
Take an arbitrary element a from R that is not 0.
Consider the principal ideal generated by a, which is aR. Since a is not 0, aR is a non-zero module over R.
Assuming every finitely generated module is free, aR must be a free R-module.
aR being free implies there exists an element b in R such that ab=1, making a a unit.
Every non-zero element in R being a unit means that R is a division ring.
Since R is also commutative, it is not merely a division ring but a field.
If there are no nonzero elements, then R can be nothing but the zero ring.
Therefore, only a field or the zero ring can satisfy the given condition, proving our initial statement.