Difference between revisions of "Ring"

Revision as of 15:00, 13 June 2008

A ring is a structure of abstract algebra, similar to a group or a field. A ring $R$ is a set of elements closed under two operations, usually called multiplication and addition and denoted $\cdot$ and $+$ , for which

$(R,+)$ is an abelian group;
$(R,\cdot)$ is a monoid;
Multiplication distributes doubly over addition.

In other words, the following properties hold for all $a,b,c$ in $R$ :

$(a+b) + c = a+(b+c)$ (associativity of addition);
$a+b = b+a$ (commutativity of addition);
For some $0\in R$ , $0+a=a+0=a$ (existance of additive identity);
There exists some $-a\in R$ for which $a+ (-a) = (-a)+a = 0$ (existance of additive inverses);
$(ab)c = a(bc)$ (associativity of multiplication);
For some $1\in R$ , $1a=a1=a$ (existance of multiplicative identity)
$\begin{align*} a(b+c)&= ab+bc \\ (b+c)a &= ba + bc \end{align*}$ (Error compiling LaTeX. Unknown error_msg) (double distributivity of multiplication over addition).

Note especially that multiplicative inverses need not exist and that multiplication need not be commutative.

The elements of $R$ under addition is called the additive group of $R$ ; it is sometimes denoted $R^+$ . (However, this can sometimes lead to confusion when $R$ is also an ordered set.) The set of invertible elements of $R$ constitute a group under multiplication, denoted $R^*$ . The elements of $R$ under the multiplicative law $(a,b) \mapsto ba$ (i.e., the opposite multiplicative law) and the same additive law constitute the opposite ring of $R$ , which can be denoted $R^0$ .

Let $a$ be an element of $R$ . Then the mapping $x \mapsto ax$ of $R$ into $R$ is an endomorphism of the abelian group $R^+$ . Since group homomorphisms map identities to identities, it follows that $a0 = 0$ , for all $a$ in $R$ , and similarly, $0a = 0$ .

Divisors

Let $x$ and $y$ be elements of a ring $R$ . If there exists an element $a$ of $R$ such that $x=ay$ , then $y$ is said to be a right divisor of $x$ , and $x$ is said to be a left multiple of $y$ . Left divisors and right multiples are defined similarly. When $R$ is commutative, we say simply that $y$ is a divisor of $x$ , or $y$ divides $x$ , or $x$ is a multiple of $y$ .

Note that the relation " $y$ is a right divisor of $x$ " is transitive, for if $x = ay$ and $y = bz$ , then $x= (ab)z$ . Furthermore, every element of $R$ is a right divisor of itself. Therefore $R$ has the (sometimes trivial) structure of a partially ordered set.

Under these definitions, every element of $R$ is a left and right divisor of 0. However, by abuse of language, we usually only call an element $x$ a left (or right) divisor of zero (or left, right zero divisors) if there is a non-zero element $y$ for which $xy=0$ (or $yx=0$ ). The left zero divisors are precisely those $x$ elements of $R$ for which left multiplication is not cancellable. For if $y,z$ are distinct elements of $R$ for which $xy=xz$ , then $x(y-z)=0$ .

Examples of Rings

The sets of integers ( $\mathbb{Z}$ ), rational numbers ( $\mathbb{Q}$ ), real numbers ( $\mathbb{R}$ ), and complex numbers ( $\mathbb{C}$ ) are all examples of commutative rings, as is the set of Gaussian integers ( $\mathbb{Z}[i]$ ). Note that of these, the integers and Gaussian integers do not have inverses; the rest do, and therefore also constitute examples of fields. All these rings are infinite, as well.

Among the finite commutative rings are sets of integers mod $m$ ( $\mathbb{Z}/m\mathbb{Z}$ ), for any integer $m$ .

If $G$ is an abelian group, then the set of endomorphisms on $G$ form a ring, under the rules $(f+g)(x) = f(x)+ g(x); \qquad fg = f\circ g .$

Let $R$ be a ring. The set of polynomials in $R$ is also a ring.

Let $F$ be a field. The set of $n\times n$ matrices of $F$ constitute a ring. In fact, they are the endomorphism ring of the additive group $(F^+)^n$ .

If $R,R'$ are rings, then Cartesian product $R_1 \times R_2$ is a ring under coordinatewise multiplication and addition; this is called the direct product of these rings.

Let $\mathcal{F}$ be the set of weak multiplicative functions mapping the positive integers into themselves. Then the elements of $\mathcal{F}$ form a pseudo-ring, with multiplication defined as Dirichlet convolution, i.e., $(fg)(n) = \sum_{d\mid n} f(d)g(n/d) ,$ for $((fg)h)(n) = (f(gh))(n) = \sum_{abc=n} f(a)f(b)f(c) .$ However, there is no multiplicative identity, so this is not a proper ring.

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-A '''ring''' is a structure of [[abstract algebra]], similar to a [[group]] or a [[field]].  A ring <math>R</math> is a [[set]] of elements with two [[operation]]s, usually called multiplication and addition and denoted <math>\cdot</math> and <math>+</math>, which have the following properties:
+A '''ring''' is a structure of [[abstract algebra]], similar to a [[group]] or a [[field]].  A ring <math>R</math> is a [[set]] of elements closed under two [[operation]]s, usually called multiplication and addition and denoted <math>\cdot</math> and <math>+</math>, for which
+* <math>(R,+)</math> is an [[abelian group]];
+* <math>(R,\cdot)</math> is a [[monoid]];
+* Multiplication distributes doubly over addition.
+In other words, the following properties hold for all <math>a,b,c</math> in <math>R</math>:
+* <math>(a+b) + c = a+(b+c)</math> (associativity of addition);
+* <math>a+b = b+a</math> (commutativity of addition);
+* For some <math>0\in R</math>, <math>0+a=a+0=a</math> (existance of additive identity);
+* There exists some <math>-a\in R</math> for which <math>a+ (-a) = (-a)+a = 0</math> (existance of additive inverses);
+* <math>(ab)c = a(bc)</math> (associativity of multiplication);
+* For some <math>1\in R</math>, <math>1a=a1=a</math> (existance of multiplicative identity)
+* <math>\begin{align*} a(b+c)&= ab+bc \\ (b+c)a &= ba + bc \end{align*}</math> (double [[distributive property |distributivity]] of multiplication over addition).
-Under the operation +, the ring is an [[abelian group]] and so obeys all the group axioms (existence of an [[identity]], existence of [[inverse with respect to an operation | inverses]], [[associative | associativity]]) as well as [[commutative | commutivity]].
+Note especially that multiplicative inverses need not exist and that multiplication need not be commutative.
+The elements of <math>R</math> under addition is called the ''additive group of <math>R</math>''; it is sometimes denoted <math>R^+</math>.  (However, this can sometimes lead to confusion when <math>R</math> is also an [[ordered set]].)  The set of invertible elements of <math>R</math> constitute a group under multiplication, denoted <math>R^*</math>.  The elements of <math>R</math> under the multiplicative law <math>(a,b) \mapsto ba</math> (i.e., the opposite multiplicative law) and the same additive law constitute the ''opposite ring of <math>R</math>'', which can be denoted <math>R^0</math>.
+Let <math>a</math> be an element of <math>R</math>.  Then the mapping <math>x \mapsto ax</math> of <math>R</math> into <math>R</math> is an [[endomorphism]] of the [[abelian group]] <math>R^+</math>.  Since group homomorphisms map identities to identities, it follows that <math>a0 = 0</math>, for all <math>a</math> in <math>R</math>, and similarly, <math>0a = 0</math>.
+== Divisors ==
-There exists an element, usually denoted 1, such that <math>1 \cdot a = a \cdot 1 = a</math> for all <math>a\in R</math>.  (Multiplicative identity.)
+Let <math>x</math> and <math>y</math> be elements of a ring <math>R</math>.  If there exists an element <math>a</math> of <math>R</math> such that <math>x=ay</math>, then <math>y</math> is said to be a ''right [[divisor]] of <math>x</math>'', and <math>x</math> is said to be a ''left [[multiple]] of <math>y</math>''.  Left divisors and right multiples are defined similarly.  When <math>R</math> is commutative, we say simply that <math>y</math> is a divisor of <math>x</math>, or <math>y</math> divides <math>x</math>, or <math>x</math> is a multiple of <math>y</math>.
-For every three elements <math>a, b, c\in R</math> we have <math>a\cdot(b\cdot c) = (a\cdot b)\cdot c</math>.  (Associativity.)
+Note that the relation "<math>y</math> is a right divisor of <math>x</math>" is [[transitive]], for if <math>x = ay</math> and <math>y = bz</math>, then <math>x= (ab)z</math>.  Furthermore, every element of <math>R</math> is a right divisor of itself.  Therefore <math>R</math> has the (sometimes trivial) structure of a [[partially ordered set]].
-For every three elements <math>a, b, c\in R</math> we have <math>a\cdot(b + c) = (a\cdot b) + (a\cdot c)</math> and <math>(a + b)\cdot c = (a\cdot c) + (b\cdot c)</math>.  ([[Distributive property | Distributivity]] of multiplication over addition.)
+Under these definitions, every element of <math>R</math> is a left and right divisor of 0.  However, by abuse of language, we usually only call an element <math>x</math> a left (or right) divisor of zero (or left, right [[zero divisor]]s) if there is a ''non-zero'' element <math>y</math> for which <math>xy=0</math> (or <math>yx=0</math>).  The left zero divisors are precisely those <math>x</math> elements of <math>R</math> for which left multiplication is not cancellable.  For if <math>y,z</math> are distinct elements of <math>R</math> for which <math>xy=xz</math>, then <math>x(y-z)=0</math>.
+== Examples of Rings ==
-Note especially that multiplicative inverses need not exist and that multiplication need not be commutative.
+The sets of [[integer]]s (<math>\mathbb{Z}</math>), [[rational number]]s (<math>\mathbb{Q}</math>), [[real number]]s (<math>\mathbb{R}</math>), and [[complex number]]s (<math>\mathbb{C}</math>) are all examples of commutative rings, as is the set of [[Gaussian integer]]s (<math>\mathbb{Z}[i]</math>).  Note that of these, the integers and Gaussian integers do not have inverses; the rest do, and therefore also constitute examples of [[field]]s.  All these rings are [[infinite]], as well.
+Among the finite commutative rings are sets of integers mod <math>m</math> (<math>\mathbb{Z}/m\mathbb{Z}</math>), for any integer <math>m</math>.
+If <math>G</math> is an abelian group, then the set of [[endomorphism]]s on <math>G</math> form a ring, under the rules
+<cmath> (f+g)(x) = f(x)+ g(x); \qquad fg = f\circ g . </cmath>
+Let <math>R</math> be a ring.  The set of [[polynomial]]s in <math>R</math> is also a ring.
+Let <math>F</math> be a field.  The set of <math>n\times n</math> [[matrices]] of <math>F</math> constitute a ring.  In fact, they are the endomorphism ring of the additive group <math>(F^+)^n</math>.
-Common examples of rings include the [[integer]]s or the integers taken [[modular arithmetic|modulo]] <math>n</math>, with addition and multiplication as usual.  In addition, every field is a ring.
+If <math>R,R'</math> are rings, then [[Cartesian product]] <math>R_1 \times R_2</math> is a ring under coordinatewise multiplication and addition; this is called the [[direct product]] of these rings.
+Let <math>\mathcal{F}</math> be the set of weak [[multiplicative function]]s mapping the positive [[integer]]s into themselves.  Then the elements of <math>\mathcal{F}</math> form a [[pseudo-ring]], with multiplication defined as [[Dirichlet convolution]], i.e.,
+<cmath> (fg)(n) = \sum_{d\mid n} f(d)g(n/d) , </cmath>
+for
+<cmath> ((fg)h)(n) = (f(gh))(n) = \sum_{abc=n} f(a)f(b)f(c) . </cmath>
+However, there is no multiplicative identity, so this is not a proper ring.
 ==See also==
+* [[Pseudo-ring]]
+* [[Ideal]]
 * [[Ring theory]]
-{{stub}}
-{{wikify}}
 [[Category:Ring theory]]

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Difference between revisions of "Ring"

Revision as of 15:00, 13 June 2008

Divisors

Examples of Rings

See also