# Ring theory

 Ring affects the special fields mathematics is special case enclosure module as special cases commutative ring (axiom K) unitary ring (N) inclined body (NEVER) body (KNEEL) zero-divisor-free ring (0) Integritätsbereich (0K) whole numbers of Z Euclidean ring main ideal ring [?] Polynomial rings of remainder class rings

the ring theory is a subsection of algebra, which concerns itself with the characteristics of rings. A ring is similar an algebraic structure, in that, as in the whole numbers [itex] \ mathbb {Z}< /math>, Addition and Multiplikation defined and with one another concerning clammy setting are compatible.

The naming ring refers not to somewhat descriptive round, but to a union from elements to a whole one. This word meaning is otherwise to a large extent lost in the German languagegone. Some older association designations (like z. B.German ring, white ring) or expressions like „criminal ring “still refer to this meaning.

The hierarchy of algebraic structures gives a view, where rings are to be arranged within the algebraic structures.

## definition (ring)

a quantity [itex] R< /math> with two binary linkages “+” and”< math> \ /math< “>cdot on [itex] R< /math> is a ring, if applies:

• [itex] (R, +)< /math> a abelsche group is, their neutral element with [itex] 0< /math> one designates;
• the multiplication “·” is associative;
• the distributive laws , D apply. h. for all [itex] A, b, C \ in R< /math> is
[itex] A \ cdot (b+c) = A \ cdot b + A \ cdot C< /math>and [itex] (a+b) \ cdot C = A \ cdot C + b \ cdot C< /math>.

## mathematical attributes for rings

unitarily; Ring with unity
a unitary ring or ring with unity is additional a ring, that a neutral element of the multiplicationpossesses. D. h. there is an element [itex] 1< /math> to math <1> \ cdot A = A applies \ cdot 1 = A< /math> for all [itex] A \ in R< /math>. For a unitary ring the above-mentioned axioms are not independent, because the Kommutativität of the addition follows outthe remaining characteristics and the existence of a one element. See in addition weakenings of the axioms.
commutatively
with a commutative ring also the multiplication is commutative. With commutative rings with unity commutative algebra is occupied.
zero divisor-free
in a zero-divisor-freeRing does not give it of [itex] 0< /math> different elements [itex] A, b< /math>, so that [itex] A cdot \ b = 0< /math>.

## characteristics

• the general feasibility of subtraction results from the group axioms of the addition.
• Each ring R is a module over itself(with itself as underlying ring). The ideals in the ring R are the straight Untermoduln of this module R.
• All multiplicatively invertable elements form the group of units.

## special cases

boolean ring
boolean rings possess as linkages andand OR operation. They arise in boolean algebra.
Body
a body is a commutative ring with unity, in that it to all [itex] A \ in R \ set minus \ {0 \}< /math> a multiplicative inverse gives.
Integrity ring
an integrity ring is a zero-divisor-free commutative ring with unity.
Local ring
a local ring is a ring with unity, in which there is exactly a maximum (left or on the right of) ideal.
Inclined body
an inclined body is a ring with unity, in that it to all [itex] A \ in R \ set minus \ {0 \}< /math> a multiplicative inverse gives.

## weakenings of the axioms

• the Kommutativität of the addition would not have for unitary rings in the definition not to be demanded, because them follow from the remaining ring axioms. For all [itex] A, b \ in R< /math> applies:
[itex] a+a+b+b=1 \ cdot a+1 \ cdot a+1 \ cdot b+1 \ cdot b= (1+1) a+ (1+1) b= (1+1) (a+b) =< /math>

[itex] 1 (a+b) +1 (a+b) =a+b+a+b< /math>
Adds one theseEquation from left with [itex] (- A)< /math> and from right with [itex] (- b)< /math>, then one receives:
[itex] a+b = b+a< /math>
Altogether with exception of the associative law of the multiplication all axioms of a unitary ring were used. The argumentation is valid thus also for non-associative unitary rings.
• One demands from [itex] (R, +)< /math> instead of the structure for the abelschen group only a commutative Monoids, and math <A> applies \ for cdot0=0 \ cdot a=0< /math> for all [itex], then one speaks< A> \ in R /math of a half ring. The used definitions differ also here, sometimes become onlya half-group, sometimes the existence of a neutral element of the multiplication demanded.
• Another Verallgemeinerung of the ring term is the nearly ring: For this only one of the two distributive laws is demanded, and the addition is not presupposed as commutative.

## literature

Over rings one reads generally something in books to algebra.