Universal side divisor

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Template:Curing-element property

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