Question

Prove that an element of an integral domain is a unit iff it generates the domain.?

Answer 1

The assertion is false.

Explanation:

Consider the ring of numbers of the form:

`a+bsqrt(2)`

where `a, b in QQ`

This is a commutative ring with multiplicative identity `1 != 0` and no zero divisors. That is, it is an integral domain. In fact it is also a field since any non-zero element has a multiplicative inverse.

The multiplicative inverse of a non-zero element of the form:

`a+bsqrt(2)" "` is `" "a/(a^2-2b^2)-b/(a^2-2b^2)sqrt(2)`.

Then any non-zero rational number is a unit, but does not generate the whole ring, since the subring generated by it will contain only rational numbers.