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.

Facts

References

Textbook references

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