Question

If any two non zero elements of R are associate then R is a field.

Short answer

It is proved that any two non-zero elements of Rare associate then, R is a field.

Step-by-step solution

Step-by-step-solutionStep 1: Use Definition of Associate

By using the definition of associate, we can say that let R be a commutative ring with unit element. Two elements a and b in R is said to be associate if where u is unit in R.

Let $\mathrm{a},\mathrm{b}\in \mathrm{R}$are associate $\Rightarrow a=bu$

Here, u is unit.

Therefore,u^-1 exist

Prove that R is a field.

We know that $a$is an associate of $b$if$b$ is an associate of$a$.

Also, a non-zero element of $R$is divisible by each of its associates.

Therefore,$\frac{a}{b}\in R$

Multiplicative inverse of$a$ exists.

Since, a is arbitrary.

Hence, R is a field.