Ring (mathematics)

In abstract algebra, a ring is an algebraic system made up of a set and two internal operations, usually called «sum» and «product», which fulfill certain properties.

In more specific terms, a ring is a tender (R,+,⋅ ⋅ ){displaystyle (R,+,cdot)}Where R{displaystyle R} is a set and + and • are internal binary operations R{displaystyle R}Where (R,+){displaystyle (R,+)} It's an Abelian group, (R,⋅ ⋅ ){displaystyle (R,cdot)} is a monoid and the bilateral distribution of • regarding + is verified. It is often called "suma" and "product" to operations + and • respectively. In this convention, the neutral element of the sum is designated as 0, the opposite with respect to the sum of an element abelonging to the whole R given, it denotes as -a and product neutral is designated as 1. It would be redundant to say that a ring is a non-empty set, since once it is defined as an abelian group with the sum, this is clear.

The product in a ring does not necessarily have a definite inverse operation, unlike other algebraic structures such as the field. If the product is commutative, such a ring is called a "commutative ring."

History

The theory of rings arose from the exploration of issues related to the divisibility among integers, from the simultaneous study of the divisibility of polynomials, and even from the case of fields, specifically, of rational numbers, real numbers, complex numbers, and of algebraic numbers, quaternions, rational fractions and others. In the initial stage, it was the subjects of number theory and algebraic geometry that led to the concepts of ring, body and ideal. In their axiomatic structuring such ideas were the fruit of the efforts of Dedekind and other mathematicians at the end of the 19th century. Its applications to mathematical analysis show the modern approaches to algebrization of such a mathematical discipline, which only occurred in the second quarter of the XX century..

The term ring was proposed by the German mathematician David Hilbert in Der Zahlbericht (Report on Numbers 1897). The phrase Boolean ring belongs to the British mathematician Arthur Harold Stone (1938).

Notion of ring

Consider the set of integers:

... –4, –3, –2, –1, 0, 1, 2, 3, 4,...

provided with two of the binary operations: addition and multiplication. Historically, the set ℤ of integers with their two operations served as the basis for the formulation of the concept of ring^{[citation needed]}. The reason why integers form a ring is that they have the following properties:

1. The integers are closed under the sum: given two integers a and b, it's fulfilled a + b It's an integer.
2. The sum is associative: given three integers a, b and c, it is fulfilled that (a + b+ c = a + (b + c).
3. There is a neutral element for the sum: for all integers a, a + 0 = 0 + a = a.
4. There is a symmetrical element for the sum: for all integers aThere's always a whole number. b, such that a + b = 0.
5. The sum is commutative: given two integers a and b, it's fulfilled a + b = b + a.
6. Integers are closed under multiplication: given two integers a and b, it's fulfilled a × b It's an integer.
7. The multiplication is associative: given three integers a, b and c, it is fulfilled that (a × b) × c = a ×b × c).
8. There is a neutral element for multiplication: for all integers a, a × 1 = a.
9. The multiplication is distributive to the sum: a ×b + c) = (a × b) + (a × c).
10. The multiplication is distributive to the sum: (b + c) × a =b × a) + (c × a).

Definition

Sea R{displaystyle R} an unempty set, and be ⋆ ⋆ {displaystyle star } and {displaystyle circ } two binary operations R{displaystyle R}. It is said that the whole (R,⋆ ⋆ , ){displaystyle (R,starcirc),} It's a group when the following properties are fulfilled:

1.	R{displaystyle R} is closed under the operation ⋆ ⋆ {displaystyle star }.	Русский Русский a,b한 한 R,a⋆ ⋆ b한 한 R{displaystyle forall a,bin R,astar bin R}	Magma
2.	The operation ⋆ ⋆ {displaystyle star } It's associative.	Русский Русский a,b,c한 한 R,(a⋆ ⋆ b)⋆ ⋆ c=a⋆ ⋆ (b⋆ ⋆ c){displaystyle forall a,b,cin R,(astar b)star c=astar (bstar c)}	Semi-group
3.	The operation ⋆ ⋆ {displaystyle star } has a neutral element n{displaystyle n}.	consuming consuming n한 한 R:Русский Русский a한 한 R,a⋆ ⋆ n=n⋆ ⋆ a=a{displaystyle exists nin R:forall ain R,astar n=nstar a=a}	Monoid
4.	There is always a symmetrical element in respect n{displaystyle n} for ⋆ ⋆ {displaystyle star }.	Русский Русский a한 한 R,consuming consuming b한 한 R:a⋆ ⋆ b=b⋆ ⋆ a=n{displaystyle forall ain R,exists bin R:astar b=bstar a=n}	Group

A fifth condition defines an abelian group:

5.

The operation ⋆ ⋆ {displaystyle star } It's commutative.

Русский Русский a,b한 한 R,a⋆ ⋆ b=b⋆ ⋆ a{displaystyle forall a,bin R,astar b=bstar a}

To define a ring, it is necessary to add four more conditions, which concern the second binary operation:

6.	R is closed under the operation {displaystyle circ }.	Русский Русский a,b한 한 R,a b한 한 R{displaystyle forall a,bin R,acirc bin R}
7.	The operation {displaystyle circ } It's associative.	Русский Русский a,b,c한 한 R,(a b) c=a (b c){displaystyle forall a,b,cin R,(acirc b)circ c=acirc (bcirc c)}
8.	The operation {displaystyle circ } has a neutral element m{displaystyle m}.	consuming consuming m한 한 R:Русский Русский a한 한 R,a m=m a=a{displaystyle exists min R:forall ain R,acirc m=mcirc a=a}
9.	The operation {displaystyle circ } is distributive to ⋆ ⋆ {displaystyle star }.	Русский Русский a,b,c한 한 R,{a (b⋆ ⋆ c)=(a b)⋆ ⋆ (a c)(a⋆ ⋆ b) c=(a c)⋆ ⋆ (b c){displaystyle forall a,b,cin R,quad left{{begin{array}{l}acirc (bstar c)=(acirc b)star (acirc c)star{end}{ !

And adding a ninth condition, a commutative ring is defined:

10.

The operation {displaystyle circ } It's commutative.

Русский Русский a,b한 한 R,a b=b a{displaystyle forall a,bin R,acirc b=bcirc a}

When the existence of a neutral of the second operation is not required, we speak of a pseudo-ring. There is also the definition of a ring that does not include the existence of a neutral element for the second operation, and in this case, the rings that do have said neutral element for the second operation and where said element is different from the neutral of the first operation.

Examples

• The whole of Gaussian integers R={m+ni:m,n한 한 Z!{displaystyle R={m+ni:m,nin mathbb {Z} }with the usual addition and multipleization is a ring. It is a subanillo of complex numbers C{displaystyle mathbb {C} }.
• The whole M{displaystyle M} of the royal matrices of order 2{displaystyle 2} with the addition and multiplication of matrices is a non-commutative ring.
• The whole Q(3){displaystyle mathbb {Q} ({scriptstyle {sqrt {3}}}}} of the real numbers: m+n3{displaystyle m+nscriptstyle {sqrt {3}}} where m,n한 한 Q{displaystyle m,nin mathbb {Q} } (they are rational), with the addition and multiplication, is a commutative ring.
• The whole Z6{displaystyle mathbb {Z} _{6}} of the entire modules 6{displaystyle 6}; with modular addition and multiplication, it is a finite ring with 0 dividers.
• The whole F[chuckles]x]{displaystyle F[x]} of polynomials with coefficients in Z{displaystyle mathbb {Z} } (together of the integers), with the addition and multiplication, is a unitary ring.

Subtraction

An operation linked to addition can be defined in a ring: subtraction.

• The difference of a and b is defined as d = a +(-b), result guaranteed by the existence and uniqueness of the opposite b. The operation you ordered to pair, b assigns your difference is called subtraction. And it is considered reverse operation of the addendum in the sense in which Русский Русский a,b한 한 R,a=d+b{displaystyle forall a,bin R,a=d+b}what we can see easily because d+b=(a+(− − b))+b=a+((− − b)+b)=a+0=a{displaystyle d+b=(a+(-b)))+b=a+(-b)+b)=a+0=a}. The subtraction solves the equation b+x = unlike a and b.

Distributive with subtraction

Русский Русский a,b,c한 한 R,a(b− − c)=ab− − ac,(b− − c)a=ba− − ca{displaystyle forall a,b,cin R,a(b-c)=ab-ac,(b-c)a=ba-ca}

Featured elements in a ring

• Element zero, denoted by 0{displaystyle}, is the neutral element for the sum. For this element the following is verified:

Be R an arbitrary ring. 0⋅ ⋅ x=0Русский Русский x한 한 R{displaystyle 0cdot x=0qquad forall xin R}

Demonstration

0x=(0+0)x=0x+0x⇒ ⇒ 0x=0x+0x{displaystyle 0x=(0+0)x=0x+0xRightarrow 0x=0x+0x} Add the reverse additive 0x{displaystyle 0x}which exists since R is a group for the sum, 0x− − 0x=0x{displaystyle 0x-0x=0x} But 0x− − 0x=0{displaystyle 0x-0x=0}. Finally 0=0xРусский Русский x한 한 A{displaystyle 0=0xqquad forall xin A}

• Multiple for any positive integer n{displaystyle n} and the element a{displaystyle a} of the ring is defined na=a+...+a.(nveces){displaystyle na=a+...+a.{} and na{displaystyle na} It's called multiple to. It is also fulfilled that 0a=0R{displaystyle 0a=0_{R}}. So the zero integer for any element of a ring is equal to zero of the ring. Finally, n(− − a)=− − na{displaystyle n(-a)=-na} where n{displaystyle n} is positive integer and − − a{displaystyle} It's him. opposite of a{displaystyle a}.

• Unit element: if an element, which we denote 1, fulfills 1⋅ ⋅ a=a⋅ ⋅ 1=a{displaystyle 1cdot a=acdot 1=a} for all elements a of the ring, it is called unitary element. The zero element and the unitary element (case of existence) only coincide in the case that the ring is trivial:

Demonstration

Sea a한 한 A:a=a⋅ ⋅ 1=a⋅ ⋅ 0=0{displaystyle ain A:a=acdot 1=acdot 0=0} Then, Русский Русский a한 한 Aa=0{displaystyle forall ain Aquad a=0}

• Inverse multiplier: in a unit ring, multipliative reverse elements can be defined as follows:
  • element b{displaystyle b} That's it. reverse multiplier on the left (or simply) reversed by the left) a{displaystyle a} Yeah. b⋅ ⋅ a=1{displaystyle bcdot a=1}.
  • Likewise, the element c{displaystyle c} That's it. reverse multiplier on the right (or simply) reversed by the right) a{displaystyle a} Yeah. a⋅ ⋅ c=1{displaystyle acdot c=1}.

Not all the elements have inverse, and it is even possible that an element has inverse on the left but not on the right, or vice versa. However, when an element a has an inverse element on the left and on the right, then both are equal, and denotes simply as an inverse element (a− − 1{displaystyle a^{-1}).

• Inversible element, invertible element or unit: is all that element that possesses inverse multiplication.

• Zero divider: an element aI was. I was. 0{displaystyle aneq 0} is divisor from zero on the left, if any bI was. I was. 0{displaystyle bneq 0}, so a·b=0. It is on the right if there is a cI was. I was. 0{displaystyle cneq 0} different from 0 such that c·a=0. It'll be said that a It is a zero splitter if it is both on the right and on the left.

• Regular element: an element aI was. I was. 0{displaystyle aneq 0} of a ring is regular if it is not divisor of zero. Every invertible element is regular.

• Idempotent element: is any element e{displaystyle e} of the ring that by multiplying itself does not vary, that is, e⋅ ⋅ e=e{displaystyle ecdot e=e} (or alternatively) e2=e{displaystyle e^{2}=e}). The zero is always idempotent in a ring, and if the ring is unitary, also the 1 is idempotent.

• Nilpotent element (o) nihilpotente): is any element x{displaystyle x} of the ring for which there is a natural number n{displaystyle n} so that xn=0{displaystyle x^{n}=0} (where) xn{displaystyle x^{n}} is defined by recurrence: x0=1{displaystyle x^{0}=1}, xn=x⋅ ⋅ xn− − 1{displaystyle x^{n}=xcdot x^{n-1}}}). 0 is always a nilpotent of any ring. Every nilpotent element is a zero divider.

Some Important Types of Rings

• Switch ring: the one in which the product is commutative, that is, a·b=b·a for all a and b (not to be confused with Abelian ring). As an example: the set P{displaystyle P} of the integers pairs with the sum and product of integers is a non-unitary commutative ring.
• Non-commutative ring is that in which the product is not commutative. For example, the whole M2× × 2{displaystyle {mathcal {M}}_{2times 2}}} of the square royal matrices 2{displaystyle 2}, with the sum and product of matrices is a non-commutative unit ring.

• Unit ring: the one who possesses a unitary element and also, this is different from the neutral of the sum (this distinction is made when it is not considered that a ring has to have a neutral element regarding the product.

• Division ring: is the ring in which every element, except for 0{displaystyle}It's inverse.

• Ring with simplifying laws: the one in which the laws of simplification are fulfilled. If a ring has no divisors of the zero, the laws of simplification are met, and the reciprocal is also true.

• Domain of integrity: if a ring does not have divisors of the zero, it is a domain of integrity (it is often required that it is also commutative and unitary rings, but this requirement is not accepted by all authors).

• Body: it is a commutative division ring.

• Abelian ring: is a ring in which every idempotent element belongs to the center of the ring, that is, every idempotent element conmutes with any element of the ring.

• Euclide ringor Euclide domain is a domain of integrity R along with an euclide N standard. The ring of the integers, the whole gaussians and the rings of polynomials are examples of eucliding domains.

• Ring integrally closed: an integral domain R{displaystyle R} is an integrally closed ring if your integral lock in your fraction field is R{displaystyle R} Right. I mean, yeah. b{displaystyle b} is an element of Frac(R{displaystyle R}) which is solution of a non-constant polynomial bn+an− − 1bn− − 1+ +a1b+a0=0,{displaystyle b^{n}+a_{n-1}b^{n-1}+cdots +a_{1b+a_{0}=0,} coefficients ai{displaystyle a_{i}} in R{displaystyle R}, then b{displaystyle b} It's in. R{displaystyle R}.

Notable Subsystems

Subrings

A subanillo S{displaystyle S} of a ring (R,+,⋅ ⋅ ){displaystyle (R,+,cdot)} It's a subset. S R{displaystyle Ssubset R} with the laws of internal composition of the ring R{displaystyle R} fulfills that, yes a,b한 한 S{displaystyle a,bin S}, then a+(− − b)한 한 S{displaystyle a+(-b)in S} and a⋅ ⋅ b한 한 S{displaystyle acdot bin S}. Yeah. 1한 한 R{displaystyle 1in R} (i.e., if the ring is unitary), then it will also be required that 1한 한 S{displaystyle 1in S}. Note that in this case, when the ring is unitary, {0} it will not be linen R{displaystyle R}and it will be if R{displaystyle R} It's not unitary.

A subanillo S{displaystyle S} it's proper when it doesn't match the whole ring, that is, if RI was. I was. S{displaystyle Rneq S}.

It turns out that a subanillo is a ring within another ring (for the same operations). In particular, (S,+){displaystyle (S,+)} is a subgroup of (R,+){displaystyle (R,+)}.

Examples:

1. Z{displaystyle mathbb {Z} } It's a subanillo. Q{displaystyle mathbb {Q} }in the same way, Q{displaystyle mathbb {Q} } It's a subanillo. R{displaystyle mathbb {R} }and R{displaystyle mathbb {R} } It's a subanillo. C{displaystyle mathbb {C} }.
2. The complex algebraic numbers set is a subanillo C{displaystyle mathbb {C} }.

Proposition

A subset K{displaystyle K} of a ring R{displaystyle R} It's subanillo. R{displaystyle R} Yes and only if

1. K{displaystyle K} is additive subgroup R{displaystyle R}.
2. Русский Русский x,and한 한 K{displaystyle forall x,yin K}, xand한 한 K{displaystyle xyin K}.

Ideals

Of much greater interest in ring theory are ideals, since they are not only closed with respect to multiplication with respect to elements of the ideal, but also when an element of the ideal is multiplied by any element of the ring:

• A subset I R{displaystyle Isubset R} That's it. ideal on the left of a ring (R,+,⋅ ⋅ ){displaystyle (R,+,cdot)} Yeah. (I,+){displaystyle (I,+)} is subgroup (R,+){displaystyle (R,+)} and any r한 한 R{displaystyle rin R} and x한 한 I{displaystyle xin I} You have to r⋅ ⋅ x한 한 I{displaystyle rcdot xin I}.

• A subset I R{displaystyle Isubset R} That's it. ideal on the right of a ring (R,+,⋅ ⋅ ){displaystyle (R,+,cdot)} Yeah. (I,+){displaystyle (I,+)} is subgroup (R,+){displaystyle (R,+)} and any r한 한 R{displaystyle rin R} and x한 한 I{displaystyle xin I} You have to x⋅ ⋅ r한 한 I{displaystyle xcdot rin I}.

When a subset I is right-ideal and left-ideal, it is said to be a two-sided ideal, or simply ideal. The commutative property ensures that in commutative rings every left-ideal is also a right-ideal, and every right-ideal is a left-ideal, that is, all ideals (either left or right) of a commutative ring are bilateral ideals.

An ideal does not necessarily have to be a subanillo. An ideal I{displaystyle I} it is said to be proper if it is different from the whole ring, that is, II was. I was. R{displaystyle Ineq R}.

Units

The set of invertible elements of a unit ring (R,+,⋅ ⋅ ,1R){displaystyle (R,+,cdot1_{R}}}, called units of R, forms a group regarding the multiplication of the ring, which receives the name of group of units of R, scored U(R){displaystyle U(R)}.

Yeah. I{displaystyle I} is ideal (on the left, on the right or bilateral) of a unit ring R{displaystyle R}, U(R){displaystyle U(R)} is the group of R units, then I U(R)=∅ ∅ {displaystyle Icap U(R)=varnothing }, that is, no own ideal has invertible elements. In particular, no ideals (on the left, on the right or bilateral) own have as an element to 1, which prevents ideals from being single rings.

For example, the units of the ring of integers are 1 and -1 (isomorphic to the group of two elements), and the group of units of square matrices of order n is the linear group general of order n, which contains matrices with a determinant other than 0.

Center

The center of a ring (R,+,⋅ ⋅ ){displaystyle (R,+,cdot)} (denoted by Z(R){displaystyle Z(R)}) is the set of elements that commut for the product, i.e. Z(R):={r한 한 R:r⋅ ⋅ s=s⋅ ⋅ r,Русский Русский s한 한 R!{displaystyle Z(R):={rin R:rcdot s=scdot r,forall sin R}}. The center of a ring comes to be like "the commutative part of the ring." Note that you always have to 0한 한 Z(R){displaystyle 0in Z(R)}. The switching rings are those that match your center, i.e., R=Z(R){displaystyle R=Z(R)}.

For example, the center of the ring of the square matrices of order n is constituted only by the scalar matrices, those that are equal to the identity matrix multiplied by a scalar.