I can't understand what actually being Associate means in Rings of Polynomials. The book states:
Two elements $a$ and $b$ of a commutative ring with unity are associates if there exists a unit $u$ such that $a=ub$. Then it says in example that in ring of polynomials ,two polynomials that are scalar multiple of one another are associates.
How did we deduce this case in polynomials from definition of Associates ? Kindly help…
Best Answer
If you are working with polynomials over a field, then any scalar is a unit.
If your polynomials are over a ring which is not a field, then there will be scalars which are not units (have no multiplicative inverse). In $\mathbb Z[x]$ it is clear that $2$ is not an associate of $1$ even though it is a scalar multiple of $1$.
I would check whether the definition in your book specifies that you are working with polynomials over a field.