Universal side divisor

Definition

A nonzero element $x$ in an commutative unital ring $R$ is termed a universal side divisor if $x$ satisfies the following two conditions:

• $x$ is not a unit.
• For any $y\in R$, either $x$ divides $y$ or there exists a unit $u\in R$ such that $x$ divides $y-u$.

Facts

• Element of smallest norm among non-units in Euclidean ring has a universal side divisor
• Euclidean ring that is not a field has a universal side divisor