Question

Question: If every non zero element of R is either irreducible or unit, prove that R is a field.

Short answer

It is proved that$R$ is a field.

Step-by-step solution

Step-by-step-solutionStep 1: If every non-zero element of R is unit.

Using definition of unit which is says that,an element $u\in R$is a unit provided that$uv=I_R$for some$v\in R$

Therefore,u^-1 exists.

Let every non-zero element of is a unit.

Therefore, for all$u\in R$ u^-1exists.

Therefore, R is a field.

If every element of R is irreducible

By using the definition of irreducible element, we can say thata non-zero element$p\in R$ is said to be irreducible provided that p is not a unit and the only divisor of p are its associates and the units of R.

Therefore, p can be written as:

$p=qu$, where q is its associate and u is its unit.

Now, every element of R is irreducible.

Therefore, by exercise 20, we can say that R is a field.