Associated elements

The associated elements of a ring are a term from the divisibility theory in mathematics . Two elements and are called associated when they are mutually divisible, ie when “ divides ” and “ divides ” are fulfilled at the same time. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle b}$ ${\displaystyle a}$

definition

Commutative rings

Two elements of an integrity ring (zero divisors free commutative ring with 1) hot mutually associated , if a unit with exists. This is fulfilled exactly when one and one another share, that is, and are fulfilled. It also writes , or . ${\displaystyle a,b}$ ${\displaystyle R}$ ${\displaystyle \epsilon }$ ${\displaystyle b=a\cdot \epsilon }$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a\mid b}$ ${\displaystyle b\mid a}$ ${\displaystyle a\sim b}$ ${\displaystyle a~{\hat {=}}~b}$

Non-commutative rings

Two elements of a non-commutative ring with one hot mutually associated right if a legal entity with exists. Then there is both the right multiple of , i.e. the left divisor of , and the right multiple of . ${\displaystyle a,b}$ ${\displaystyle R}$ ${\displaystyle \epsilon }$ ${\displaystyle b=a\cdot \epsilon }$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$

Accordingly, one defines left associated with a left unit and left multiple. If two elements are associated both left and right, they are considered to be bilaterally associated . ${\displaystyle a,b\,}$

In addition, two elements can be defined as extended associated if there are 2 units with . They are then not necessarily in a divisible relationship , but it follows from two-sided associated both left-associated and right-associated and both left-associated and right-associated and extended associated . ${\displaystyle a,b\in R}$ ${\displaystyle \delta ,\epsilon }$ ${\displaystyle b=\delta \cdot a\cdot \epsilon }$ ${\displaystyle a,b}$

Comment:

In the non-commutative case, one has to name the side (left, right) of the divisor and multiple property, which the simple divisibility symbol (whose symmetrical shape already stands in the way of a reflection with inverse meaning) of the commutative case cannot express.

properties

Association is an equivalence relation (also the three forms including the extended in the non-commutative case). It is compatible with the divisor relation (in the non-commutative case in the correctly selected laterality), that is, for laterally associated elements the divisors or multiples of are exactly the divisors or multiples of . ${\displaystyle a,b}$ ${\displaystyle a}$ ${\displaystyle b}$

In an integrity ring, two elements are associated if and only if they produce the same main ideal .

Examples

• In the ring of whole numbers are associated if and only if applies. This is because in numbers and are the only units. ${\displaystyle \mathbb {Z} }$ ${\displaystyle a,b}$ ${\displaystyle a=\pm b}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle 1}$ ${\displaystyle -1}$
• In a body all of the different elements are associated with each other. ${\displaystyle 0}$
• In polynomial over a field , two elements , and if and only associated when a exists with . ${\displaystyle K[x]}$ ${\displaystyle K}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle a\in K\setminus \{0\}}$ ${\displaystyle g=a\cdot f}$
• In a factorial ring , apart from the zero element, every non-unit has a decomposition into irreducible elements , which is unambiguous except for order and association.
• In the non-commutative ring of the Hurwitz quaternions , the group of 24 units is not commutative. Apart from the Hurwitz quaternions with norm and the purely real ones, which only have bilateral associations, the others also have unilaterally (both right and not left as well as left and not right ) associations, and a right or left ideal created by them is not bilateral. ${\displaystyle \left\{\pm 1,\pm \mathrm {i} ,\pm \mathrm {j} ,\pm \mathrm {k} ,{\tfrac {1}{2}}(\pm 1\pm \mathrm {i} \pm \mathrm {j} \pm \mathrm {k} )\right\}}$ ${\displaystyle 2}$

Individual evidence

1. ^ Günter Scheja, Uwe Storch: Textbook of Algebra. Including linear algebra: Textbook of Algebra, Part 2 . Teubner Verlag, 1988, ISBN 3-519-02212-5 , pp. 132 ( limited preview in Google Book search).