# 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. For full proof, refer: Universal side divisor property is invariant upto associates

## 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.