Question

Question: Let be a commutative ring. If
and(with ) is a zero divisor in prove that is a zero divisor in

Short answer

Answer:

Hence it is proved that $a_0$ is a zero divisor in $R$ .

Step-by-step solution

Polynomial Arithmetic:

If any given functionis a ring, then the commutative, associative, and distributive laws hold such that the function $f\left(x\right)+g\left(x\right)$ exists.

Polynomial Operations:

The given function is a zero divisor:

Now, according to the definition, we must have a function:

Such that:

Therefore, we have:

Then, for k= m+n , we get:

a_m b_{n =0}

This implies that a_m is a zero divisor. Hence proved, a_n is a zero divisor in $R$ .