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 $$y - u$$.

Equivalently, a non-zero non-unit element is a universal side divisor if and only if the unit balls centered around its multiples cover the whole ring.

Equivalence up to associate classes
If $$x,y$$ are associate elements in a commutative unital ring $$R$$, then $$x$$ is a universal side divisor if and only if $$y$$ is a universal side divisor.

Examples

 * In the ring of rational integers $$\mathbb{Z}$$, the only universal side divisors are $$\pm 2, \pm 3$$. $$2$$ is a universal side divisor because every integer is either a multiple of $$2$$ or differs by $$1$$ from a multiple of $$2$$. $$3$$ is a universal side divisor because every integer is either $$0$$, $$1$$, or $$-1$$ mod $$3$$. For any integer $$n$$ of absolute value greater than $$3$$, there is no way of subtracting a unit or zero from $$2$$ to get a multiple of $$n$$.
 * In the polynomial ring over a field, the only universal side divisors are the nonconstant linear polynomials. For instance, in the ring $$k[x]$$, $$x$$ is a universal side divisor because for any polynomial, we can subtract the constant term of the polynomial to obtain a multiple of $$x$$, and the constant term is either zero or a unit. Similar reasoning applies for all the other nonconstant linear polynomials. On the other hand, for any polynomial of degree greater than $$1$$, there is no way of subtracting a unit from $$x$$ to get a multiple of that polynomial.

Facts

 * Element of minimum norm among non-units in Euclidean ring is a universal side divisor
 * Euclidean ring that is not a field has a universal side divisor
 * Universal side divisor implies irreducible
 * Irreducible not implies universal side divisor