Question

Letbe$R$ a non-zero finite commutative ring with no zero divisors. Prove that R is a field.

Short answer

It is proved that $R$is a field.

Step-by-step solution

Statement of Exercise 39

Exercise 39states that when $R$is a finite commutative ring with identity and$a\in R$, then $a$ would be either a zero divisor or a unit.

Show that R is a field

Any non-zero element of a finite commutative ring that has an identity is either a zero divisor or a unit according to exercise 39.

Therefore, $R$is a field because it has no zero divisors and its non-zero element is a unit.

Thus, it is proved that$R$ is a field.