 Associate element

Two elements $a,b$ of a commutative ring are associates if $a=ub$ for some unit $u$. This is an equivalence relation: reflexivity uses $1$, symmetry uses $u^{-1}$, and transitivity uses a product of units.

In $\mathbb{Z}$, the units are $1,-1$, so associates differ only by sign. In the polynomial ring $k[X]$ over a field, two nonzero polynomials are associates exactly when one is a nonzero scalar multiple of the other. Unique factorization identifies factors up to this unavoidable ambiguity. In an integral domain, nonzero elements generate the same principal ideal exactly when they are associates.