Universal side divisor
From Commalg
Template:Curing-element property
Contents
Definition
A nonzero element in an commutative unital ring
is termed a universal side divisor if
satisfies the following two conditions:
-
is not a unit.
- For any
, either
divides
or there exists a unit
such that
divides
.
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 are associate elements in a commutative unital ring
, then
is a universal side divisor if and only if
is a universal side divisor. For full proof, refer: Universal side divisor property is invariant upto associates
Examples
- In the ring of rational integers
, the only universal side divisors are
.
is a universal side divisor because every integer is either a multiple of
or differs by
from a multiple of
.
is a universal side divisor because every integer is either
,
, or
mod
. For any integer
of absolute value greater than
, there is no way of subtracting a unit or zero from
to get a multiple of
.
- In the polynomial ring over a field, the only universal side divisors are the nonconstant linear polynomials. For instance, in the ring
,
is a universal side divisor because for any polynomial, we can subtract the constant term of the polynomial to obtain a multiple of
, 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
, there is no way of subtracting a unit from
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