Ring (algebra)

A ring is an algebraic structure in which, similar to the whole numbers , addition and multiplication are defined and compatible with one another in terms of brackets. The ring theory is a branch of algebra that deals with the properties of rings. ${\displaystyle \mathbb {Z} }$

Naming

The name ring does not refer to something vividly ring-shaped, but to an organized amalgamation of elements into a whole. This word meaning has otherwise largely been lost in the German language. Some older club names (such as Deutscher Ring , Weißer Ring , Maschinenring ) or expressions such as “ criminal ring ” , “ exchange ring ” or “ ring lecture ” still refer to this meaning. The concept of the ring goes back to Richard Dedekind ; however, the name ring was introduced by David Hilbert . In special situations, in addition to the term ring , the term area is also used . In the literature, you will find the term integrity area rather than integrity ring .

Definitions

Depending on the sub-area and textbook (and partly depending on the chapter) a ring is understood to mean something different. The definitions of morphisms as well as substructures and superstructures are also slightly different. In mathematical terms, these different ring terms are different categories .

ring

A ring is a set with two two- digit operations and such ${\displaystyle (R,+,\cdot )}$ ${\displaystyle R}$ ${\displaystyle +}$ ${\displaystyle \cdot }$

• ${\displaystyle (R,+)}$is an abelian group ,
• ${\displaystyle (R,\cdot )}$is a semigroup ,
• the distributive laws

${\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}$ and

${\displaystyle (a+b)\cdot c=(a\cdot c)+(b\cdot c)}$

for all are fulfilled. ${\displaystyle a,b,c\in R}$

The neutral element of is called the zero element of the ring . ${\displaystyle 0}$ ${\displaystyle (R,+)}$ ${\displaystyle R}$

A ring is called commutative if he respect to the multiplication is commutative is, otherwise it is called a non-commutative ring.

Ring with one (unitary ring)

If the semigroup has a (double-sided) neutral element , i.e. is a monoid , then one calls a ring with one or a unitary ring . Rings with only a left or only right neutral element are not considered unitary in ring theory. ${\displaystyle (R,\cdot )}$ ${\displaystyle 1}$ ${\displaystyle (R,+,\cdot )}$

Some authors basically understand a ring to be a (commutative) ring with one and otherwise speak of a pseudo ring , also known as rng (sic!) Or non-unital ring .
In the category of rings with one, the one must also be retained for ring homomorphisms.

Each ring can be embedded in a unitary ring.

→ Main article : Adjunction (one element)

Commutative ring with one

In commutative algebra , rings are defined as commutative rings with one .

Inferences

• The following applies to all : ${\displaystyle a,b\in R}$

${\displaystyle (-a)\cdot b=-(a\cdot b)=a\cdot (-b)}$

as well as "minus times minus results in plus":

${\displaystyle (-a)\cdot (-b)=a\cdot b}$.

• The addition of the additive inverse (with the unary minus ) to a ring element is called subtraction (the second from the first ring element). The operation sign for this is the binary minus sign (as opposed to the unary minus sign for the inverse formation):

${\displaystyle a-b:=a+(-b)}$.

• The distributive laws also apply to subtraction:

${\displaystyle a\cdot (b-c)=(a\cdot b)-(a\cdot c)}$,

${\displaystyle (a-b)\cdot c=(a\cdot c)-(b\cdot c)}$.

• The neutral element of addition is also the absorbing element of multiplication: ${\displaystyle 0}$

	${\displaystyle 0\cdot a}$	${\displaystyle =(0\cdot a)+0}$	(Neutrality of 0)
		${\displaystyle =(0\cdot a)+{\bigl (}(0\cdot a)-(0\cdot a){\bigr )}}$	(Property of the additive inverse)
		${\displaystyle ={\bigl (}(0\cdot a)+(0\cdot a){\bigr )}-(0\cdot a)}$	(Associativity of addition)
		${\displaystyle ={\bigl (}(0+0)\cdot a{\bigr )}-(0\cdot a)}$	(Distributive law)
		${\displaystyle =(0\cdot a)-(0\cdot a)}$	(Neutrality of 0)
		${\displaystyle =0}$	(Property of the additive inverse)

mirrored:

${\displaystyle a\cdot 0=0}$.

• If the neutral element of the multiplication coincides with that of the addition, then the ring consists of only one element. Such a ring is called a " null ring ". It is a commutative ring with one.

Lower and upper structures

Lower and upper ring

A subset of a ring is called a subring (or partial ring ) of if, together with the two restricted links of, there is again a ring. is a subring of if and only if a subgroup is with respect to addition and closed with respect to multiplication, i. H. ${\displaystyle U}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle R}$ ${\displaystyle U}$ ${\displaystyle R}$ ${\displaystyle U}$ ${\displaystyle U}$

${\displaystyle x\cdot y\in U}$if and . ${\displaystyle x\in U}$ ${\displaystyle y\in U}$

Even if a ring is a one, the one does not necessarily have to be included in. can also be a ring without a one - for example - or have another one. In the category of rings with one, a sub-ring is required to contain the same single element (for this it is necessary, but not always sufficient, that the sub-ring contains a multiplicatively neutral element in relation to it). ${\displaystyle R}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle 2\mathbb {Z} \subseteq \mathbb {Z} }$

The intersection of sub-rings is again a sub-ring, and the sub-ring produced by is defined as the average of all the comprehensive sub-rings of . ${\displaystyle A\subseteq R}$ ${\displaystyle A}$ ${\displaystyle R}$

A ring is called an upper ring or extension of a ring if a lower ring is of. It is also common to speak of a ring expansion when looking at a ring with a top ring. This is analogous to the concept of body enlargement . ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle S}$

example 1

→ Main article : Adjunction (one element)

Each ring can be embedded in a single element ring.

Example 2

The following ring expansion can be found in E. Sernesi: Deformations of algebraic schemes :
Let be a commutative ring, a module and the direct sum of the Abelian groups . A multiplication on is defined by ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle S=R\oplus M}$ ${\displaystyle S}$

${\displaystyle (a,x)\cdot (b,y)=(ab,ay+bx).}$

(The identification of with with one for which is, and the calculation of results in the formula mentioned.) Turns out to be a ring. You have the exact sequence ${\displaystyle (a,x)}$ ${\displaystyle a+\varepsilon x}$ ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon ^{2}=0}$ ${\displaystyle (a+\varepsilon x)(b+\varepsilon y)}$ ${\displaystyle S}$

${\displaystyle 0\to M\to S{\overset {p}{{}\to {}}}R\to 0}$

with the projection . Thus an extension of um . Another remarkable feature of this design is that the module becomes the ideal of a new ring . Nagata calls this process the principle of idealization . ${\displaystyle p}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle S}$

ideal

→ Main article : Ideal (ring theory)

A subset of the left ideal (or right ideal ) for a ring is called if: ${\displaystyle R}$ ${\displaystyle I}$ ${\displaystyle R}$

• ${\displaystyle I}$is a subgroup of . ${\displaystyle (R,+)}$
• For everyone and is also (or ). ${\displaystyle a\in I}$ ${\displaystyle x\in R}$ ${\displaystyle x\cdot a\in I}$ ${\displaystyle a\cdot x\in I}$

If both the left and the right ideal are both, it is called a two-sided ideal or just ideal. ${\displaystyle I}$ ${\displaystyle I}$

If a (left, right) ideal contains the one in a ring with one, it encompasses the whole . Since there is also an ideal, is the only (left, right) ideal that contains the one. and are the so-called trivial ideals. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle \lbrace 0\rbrace }$

Restricted to the subsets of , the term ideal is synonymous with the term - module , i.e. also left ideal with -link module , etc. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

Every ideal of is also a subring of , possibly without a one. In the category of rings with 1 it is not considered a sub-ring. ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle I}$

Factor ring

→ Main article : Factor ring

If there is an ideal in a ring , then one can determine the number of secondary classes ${\displaystyle I}$ ${\displaystyle R}$

${\displaystyle R/I:=\{x+I\mid x\in R\}}$

form. The link can always be continued because of its commutativity ; the connection, however, only when a two-sided ideal is in. If this is the case, then with the induced links there is a ring. It is called a factor ring - spoken: modulo . ${\displaystyle +}$ ${\displaystyle R/I}$ ${\displaystyle \cdot }$ ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle R/I}$ ${\displaystyle R/I}$ ${\displaystyle R}$ ${\displaystyle I}$

The ring homomorphism

${\displaystyle \varphi \colon R\to R/I}$,

who assigns its secondary class to an element has as its core. ${\displaystyle x}$ ${\displaystyle x+I=:{\bar {x}}}$ ${\displaystyle I}$

Base ring

In a ring with a one, the sub-ring created by is called the base ring . Did this finite thickness then the characteristics of abbreviated and is said to have positive characteristics. Otherwise it is set. This means that the unitary ring homomorphism is in the finite and infinite case ${\displaystyle R}$ ${\displaystyle 1}$ ${\displaystyle k,}$ ${\displaystyle k}$ ${\displaystyle R,}$ ${\displaystyle \operatorname {char} (R)=k,}$ ${\displaystyle R}$ ${\displaystyle \operatorname {char} (R)=0}$

${\displaystyle {\begin{array}{llll}\mathbb {Z} /\left(\operatorname {char} (R)\,\mathbb {Z} \right)&\to &R\\{\bar {n}}&\mapsto &n\cdot 1=1\cdot n\end{array}}}$

injective . The base ring is the picture and each of its elements is interchangeable with each ring element . Also is for each ring element ${\displaystyle {\bar {\mathbb {Z} }},}$ ${\displaystyle a\in R}$

${\displaystyle (-1)\cdot a=a\cdot (-1)=-a}$

the additive inverse of ${\displaystyle a.}$

Polynomial ring

→ Main article : Polynomial ring

If there is a commutative ring with one, the polynomial ring can be formed. This consists of polynomials with coefficients from and the variables together with the usual addition and multiplication for polynomials. Properties of are partially transferred to the polynomial ring. If zero divisors, factorial or Noetherian, this also applies . ${\displaystyle R}$ ${\displaystyle R[X]}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R[X]}$

Die ring

→ Main article : die ring

If a ring is one, the die ring can be formed. This consists of the square matrices with entries from with the usual addition and multiplication for matrices. The die ring is again a ring with a one. However, the matrix ring is neither commutative nor zero divisor free, even if it has these properties. ${\displaystyle R}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle R^{n\times n}}$ ${\displaystyle R}$ ${\displaystyle n>1}$ ${\displaystyle R}$

Direct product

If and are rings, then the bulk product can naturally be given a ring structure: ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R\times S}$

• ${\displaystyle (r_{1},s_{1})+(r_{2},s_{2}):=(r_{1}+r_{2},s_{1}+s_{2})}$
• ${\displaystyle (r_{1},s_{1})\cdot (r_{2},s_{2})\;\;:=(r_{1}\cdot r_{2},s_{1}\cdot s_{2})}$

Because the validity of the distributive law in every component is transferred directly to the quantity product.

If both rings are and are unitary, then is also unitary with as the one element. ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R\times S}$ ${\displaystyle (1_{R},1_{S})}$

The same construction is possible with any family of rings: If rings are over an index set , then a ring, called the direct product, is a subring of the direct product is the direct sum in which only finitely many components are different from 0. ${\displaystyle (R_{i})_{i\in I}}$ ${\displaystyle I}$ ${\displaystyle \prod _{i\in I}R_{i}}$ ${\displaystyle R_{i}.}$

Homomorphism

Ring homomorphism

→ Main article : ring homomorphism

For two rings and is called an illustration ${\displaystyle R}$ ${\displaystyle S}$

${\displaystyle \varphi \colon R\to S}$

Ring homomorphism ( homomorphism for short ), if the following applies to all : ${\displaystyle x,y\in R}$

${\displaystyle \varphi (x+y)=\varphi (x)+\varphi (y)}$       and

${\displaystyle \varphi (x\cdot y)\;\;=\varphi (x)\cdot \varphi (y).}$

The core of ring homomorphism is a two-sided ideal in . ${\displaystyle \operatorname {ker} \varphi :=\lbrace x\in R\mid \varphi (x)=0\rbrace }$ ${\displaystyle \varphi }$ ${\displaystyle R}$

A morphism of rings with one must also meet the condition that the one element is mapped onto the one element: ${\displaystyle \varphi }$

${\displaystyle \varphi (1_{R})=1_{S}}$

Isomorphism

→ Main article : Isomorphism

An isomorphism is a bijective homomorphism. The rings and are called isomorphic if there is an isomorphism from to . In this case the inverse mapping is also an isomorphism; the rings then have the same structure. ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle S}$

example

Equipped with the addition and multiplication by components, the direct product is a ring. Then with the picture ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$ ${\displaystyle r,s\in \mathbb {Z} }$

${\displaystyle {\begin{array}{llll}&\mathbb {Z} &\to &\mathbb {Z} \times \mathbb {Z} \\&z&\mapsto &(rz,sz)\end{array}}}$

a homomorphism of rings; a homomorphism of rings with one but only if ${\displaystyle (r,s)=(1,1).}$

Special elements in a ring

Divisors and zero divisors

→ Main article : Divisibility

From two elements is left divider (divider links) of , if a with exists. Then is also a right multiple of . Correspondingly, one defines right divisor ( right divider ) and left multiple . ${\displaystyle a,b\in R}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle x\in R}$ ${\displaystyle b=a\cdot x}$ ${\displaystyle b}$ ${\displaystyle a}$

In commutative rings, a left divider is also a right one and vice versa. One also writes here if is a divisor of . ${\displaystyle a\mid b}$ ${\displaystyle a}$ ${\displaystyle b}$

All elements of are (right or left) divisors of zero. The concept of the (right or left) zero divider has a different definition. If after this counts as a zero divisor, the following applies: An element is a (right or left) zero divisor if and only if it cannot be shortened (right or left) . ${\displaystyle R}$ ${\displaystyle 0}$

Invertability, unity

→ Main article : Unit (mathematics)

If there is an element in a ring with one for an element , so that (or ) holds, then one calls a left inverse (or right inverse ) of . Has both left and right inverse, it is called invertible or unity of the ring. The set of units in a ring with one is usually denoted by or . forms a group with regard to the ring multiplication - the unit group of the ring. If , then, is a skewed body , is furthermore commutative, so is a body . ${\displaystyle R}$ ${\displaystyle u}$ ${\displaystyle x}$ ${\displaystyle xu=1}$ ${\displaystyle ux=1}$ ${\displaystyle x}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle R}$ ${\displaystyle R^{*}}$ ${\displaystyle R^{\times }}$ ${\displaystyle R^{*}}$ ${\displaystyle R^{*}=R\backslash \left\{0\right\}}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

In commutative rings with one (especially integrity rings ), the units are alternatively defined as those elements that share one. That the one shares means that there is with . ${\displaystyle u}$ ${\displaystyle x}$ ${\displaystyle xu=ux=1}$

Associated elements

→ Main article : Associated elements

Two elements and are associated to the right if and only if there is a legal entity such that . Links associated with a left unit . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle u}$ ${\displaystyle au=b}$ ${\displaystyle ua=b}$ ${\displaystyle u}$

If in a commutative ring with one the elements are related to and , then and are associated with one another . The sideways (left, right) can therefore be omitted. ${\displaystyle a,b}$ ${\displaystyle a\mid b}$ ${\displaystyle b\mid a}$ ${\displaystyle a}$ ${\displaystyle b}$

Association is an equivalence relation .

Irreducibility

A different from 0 element is irreducible if it is neither left unit is still legal entity and there is no non-Left unit Right unit Not, not with out there, so if it follows from the equation that links unit or legal unit. ${\displaystyle q}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle q=ab}$ ${\displaystyle a}$ ${\displaystyle b}$

In a commutative ring it suffices to stipulate that it is different from 0 , that it is not a unit and that it follows that or is a unit. ${\displaystyle q}$ ${\displaystyle q=ab}$ ${\displaystyle a}$ ${\displaystyle b}$

Prime element

For commutative unitary rings one defines: An element is called a prime or prime element if it is not a unit and not equal to 0 and follows from or (see also main article: prime element ). ${\displaystyle p}$ ${\displaystyle p\mid ab}$ ${\displaystyle p\mid a}$ ${\displaystyle p\mid b}$

In a ring free of zero divisors, every prime element is irreducible. Conversely, in a factorial ring , every irreducible element is also a prime element.

Special cases

body

A body is a commutative ring with one, in which there is a group, i.e. a multiplicative inverse exists for each non-zero element. ${\displaystyle (R\setminus \left\{0\right\},\cdot )}$

Simple ring

A ring that is not the zero ring is called simple when the trivial ideals and the only two-sided ideals. A commutative simple ring with one is a body . ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle \{0\}}$

Idempotent ring

An idempotent ring is a ring in which the idempotence law is also fulfilled for all elements. Every idempotent ring is commutative. ${\displaystyle a\cdot a=a}$

Boolean ring

A Boolean ring is an idempotent ring with a one.

Local ring

A local ring is a ring in which there is exactly one maximum left (or right) ideal. Quite a few authors demand that a local, commutative ring must also be Noetherian and call a non-Noetherian ring with exactly one maximum ideal a quasi-local ring. In Wikipedia we omit this additional requirement and, if necessary, speak explicitly of Noetherian local rings.

Integrity ring

An integrity ring or integrity domain is a zero-divisor, commutative ring with a one that is different from zero. Every finite ring of integrity is a body. A body can be assigned to each integrity ring, the quotient body of the integrity ring.

Factorial ring, ZPE ring

A factorial ring or ZPE ring is an integrity ring in which all elements except zero have an essentially unambiguous decomposition into prime factors.

Main ideal ring

A main ideal ring is a ring of integrity in which every ideal is a main ideal . Each main ideal ring is a ZPE ring.

Euclidean ring

In a Euclidean ring there is division with remainder . This allows the greatest common divisor of two elements to be calculated using the Euclidean algorithm . Every Euclidean ring is a main ideal ring.

Noetherian Ring

In a commutative Noetherian ring all ideals are finitely generated.

Examples

• The zero ring , which consists of only one element, is a commutative ring with one ( ). ${\displaystyle 1=0}$
• The whole numbers with the usual addition and multiplication form a Euclidean ring. ${\displaystyle (\mathbb {Z} ,+,\cdot )}$
• The rational numbers with the usual addition and multiplication form a body. ${\displaystyle (\mathbb {Q} ,+,\cdot )}$
• The ring of even numbers is a commutative ring without a one. ${\displaystyle 2\mathbb {Z} }$
• Polynomial rings over a body are Euclidean rings. ${\displaystyle K[X]}$ ${\displaystyle K}$
• The matrix ring is for , ring with one, a non-commutative ring with one (the identity matrix ). ${\displaystyle R^{n\times n}}$ ${\displaystyle n>1}$ ${\displaystyle R}$
• Factor rings provide examples of rings that are not zero divisors. More precisely, for a commutative ring with one, it is true that an integrity ring is only if is a prime ideal. ${\displaystyle R/I}$ ${\displaystyle I\subseteq R}$
• The set of natural numbers with the usual addition and multiplication does not form a ring, since the addition over the natural numbers cannot be inverted. ${\displaystyle (\mathbb {N} ,+,\cdot )}$

Generalizations

Half ring

A half-ring is not an Abelian group, but only a half-group, which should also often (depending on the definition) be commutative and / or a monoid that does not have to apply to all (the definitions are not uniform). ${\displaystyle \left(H,+\right)}$ ${\displaystyle \left(H,+,0\right)}$ ${\displaystyle a\cdot 0=0\cdot a=0}$ ${\displaystyle a\in R}$

Fastring

In a fast ring , only one of the two distributive laws is required and the addition does not have to be commutative.

Alternative ring

With the alternative rings, the associativity of multiplication is dispensed with and only the alternative is required. The best-known example are the octonions , which are even an alternative body .

See also

• Algebra over a commutative ring
• Counter ring

literature

• Siegfried Bosch : Algebra . 7th edition. Springer-Verlag, Berlin / Heidelberg 2009, ISBN 978-3-540-92811-9 , doi: 10.1007 / 978-3-540-92812-6 .
• David Eisenbud : Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag, New York NY a. a. 1996, ISBN 0-387-94269-6 .
• Serge Lang : Algebra. Revised 3rd Edition. Springer-Verlag, Berlin a. a. 2002, ISBN 0-387-95385-X .
• Hideyuki Matsumura: Commutative Ring Theory. Cambridge University Press, Cambridge 1989, ISBN 0-521-36764-6 .
• Robert Wisbauer: Fundamentals of module and ring theory. A handbook for study and research. Fischer, Munich 1988, ISBN 3-88927-044-1 .

Individual evidence

1. ^ Earliest Known Uses of Some of the Words of Mathematics (R) . July 17, 2007
2. ^ The development of Ring Theory (July 17, 2007)
3. ↑ limited preview in the Google book search
4. ↑ Masayoshi Nagata : Local rings . Interscience Publishers, New York-London 1962, ISBN 0-88275-228-6 .
5. ↑ A body is called a prime body .