Associate element
= Associate element
= Associate
{synonym}
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.