Universal side divisor: Difference between revisions

Latest revision as of 15:59, 5 February 2009

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

Template:Booklink-defined

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+{{curing-element property}}
 ==Definition==
@@ Line 6: / Line 7: @@
 * For any <math>y \in R</math>, either <math>x</math> divides <math>y</math> or there exists a unit <math>u \in R</math> such that <math>x</math> divides <math>y - u</math>.
+Equivalently, a non-zero non-unit element is a universal side divisor if and only if the [[defining ingredient::unit ball in a commutative unital ring|unit balls]] centered around its multiples cover the whole ring.
+===Equivalence up to associate classes===
+If <math>x,y</math> are [[associate element]]s in a commutative unital ring <math>R</math>, then <math>x</math> is a universal side divisor if and only if <math>y</math> is a universal side divisor. {{proofat|[[Universal side divisor property is invariant upto associates]]}}
+==Examples==
+* In the [[ring of rational integers]] <math>\mathbb{Z}</math>, the only universal side divisors are <math>\pm 2, \pm 3</math>. <math>2</math> is a universal side divisor because every integer is either a multiple of <math>2</math> or differs by <math>1</math> from a multiple of <math>2</math>. <math>3</math> is a universal side divisor because every integer is either <math>0</math>, <math>1</math>, or <math>-1</math> mod <math>3</math>. For any integer <math>n</math> of absolute value greater than <math>3</math>, there is no way of subtracting a unit or zero from <math>2</math> to get a multiple of <math>n</math>.
+* In the [[polynomial ring over a field]], the only universal side divisors are the nonconstant linear polynomials. For instance, in the ring <math>k[x]</math>, <math>x</math> is a universal side divisor because for any polynomial, we can subtract the constant term of the polynomial to obtain a multiple of <math>x</math>, 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 <math>1</math>, there is no way of subtracting a unit from <math>x</math> to get a multiple of that polynomial.
 ==Facts==
-* [[Element of smallest norm among non-units in Euclidean ring is a universal side divisor]]
+* [[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]]
+==References==
+===Textbook references===
+* {{booklink-defined|DummitFoote|277|Section 8.1 (''Euclidean domains'')}}