Universal side divisor
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
- 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.
- 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