Difference between revisions of "Ring"

m
m
 
Line 10: Line 10:
 
* <math>(ab)c = a(bc)</math> (associativity of multiplication);
 
* <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)
 
* For some <math>1\in R</math>, <math>1a=a1=a</math> (existance of multiplicative identity)
*  <math>a(b+c)&= ab+ac \ (b+c)a = ba + ca = ab+ac }</math> (double [[distributive property |distributivity]] of multiplication over addition).
+
*  <math>a(b+c)= ab+ac \ (b+c)a = ba + ca = ab+ac </math> (double [[distributive property |distributivity]] of multiplication over addition).
*  <math>a(b-c)&= ab-ac \ (b-c)a = ba - ca </math> (double [[distributive property |distributivity]] of multiplication over subtraction).
+
*  <math>a(b-c)= ab-ac \ (b-c)a = ba - ca </math> (double [[distributive property |distributivity]] of multiplication over subtraction).
  
 
Note especially that multiplicative inverses need not exist and that multiplication need not be commutative.
 
Note especially that multiplicative inverses need not exist and that multiplication need not be commutative.

Latest revision as of 05:16, 8 April 2015

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

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)
  • $a(b+c)= ab+ac \\ (b+c)a = ba + ca = ab+ac$ (double distributivity of multiplication over addition).
  • $a(b-c)= ab-ac \\ (b-c)a = ba - ca$ (double distributivity of multiplication over subtraction).

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.

See also