Elements in same orbit under multiplication by group of units are associate

Statement

Suppose $R$ is a commutative unital ring and $a,b\in R$ are elements such that there exists a unit $u\in R$ such that $b=au$. Then, $a$ and $b$ are associate elements in $R$.

Definitions used

Unit

Further information: Unit An element $u$ of a commutative unital ring $R$ is termed a unit if there exists $v\in R$ such that $uv=1$.

Associate elements

Further information: Associate elements

Two elements $a,b$ of a ring $R$ are termed associate elements if they divide each other: $a|b$ and $b|a$.

Related facts

• Associate implies same orbit under multiplication by group of units in integral domain
• Associate not implies same orbit under multiplication by group of units

Proof

Given: A commutative unital ring $R$, elements $a,b\in R$ such that $b=au$ for a unit $u$ of $R$.

To prove: $a$ and $b$ are associate elements in $R$.

Proof: By the definition of unit, there exists $v\in R$ such that $uv=1$. We then have: $bv=(au)v=a(uv)=a1=a$. Thus, $b|a$. Also, $au=b$, so $a|b$. Thus, $a,b$ are associate elements.