Rings

• Definitions
  • Rings
  • unity or identity
  • commutative rings
  • division rings
• Examples

A $group$ is a set equipped with only one binary operation.But many sets are naturally endowed with two binary operations: addition and multiplication, and are denoted as usual by "+" and $\cdot$, respectively. Examples that quickly come to mind are the integers, the integers modulo n, the real numbers, matrices, and polynomials. When considering these sets as groups, we simply used addition and ignored multiplication. In many instances, however, one wishes to take into account both addition and multiplication. One abstract concept that does this is the concept of a $ring$.

Definitions

Rings

By a $ring$ we mean a set $\mathbb{A}$ with operations called addition and multiplication which satisfy the following axioms:

• $\mathbb{A}$ with additon alone is an abelian group.
• Multiplication is associative. That is, for all a,b, and c in $\mathbb{A}$, $$(a\cdot b)\cdot c=a\cdot (b\cdot c)$$
• Multiplication is distributive over addition. That is, for all a,b, and c in $\mathbb{A}$, $$a\cdot (b+c)=a\cdot b+a\cdot c$$ and $$(b+c)\cdot a=b\cdot a+c\cdot a$$

Since $\mathbb{A}$ with addition alone is an abelian group, there is in $\mathbb{A}$ a neutral element for addition: it is called the $zero$ element and is written 0. Also, every element has an additive inverse called its $negative$; the negative of $a$ is denoted by $-a$. Subtraction is defined by $$a-b=a+(-b)$$

unity or identity

If there is an element $1\in \mathbb{A}$ such that $1\ne 0$ and $1\cdot a=a\cdot 1=a$ for each element $a\in \mathbb{A}$, we say that $\mathbb{A}$ is a ring with $unity$ or $identity$.

commutative rings

A ring $\mathbb{A}$ for which $a\cdot b=b\cdot a$ for all a, b in $\mathbb{A}$ is called a $commutativering$.

division rings

A $divisionring$ is a ring $\mathbb{A}$, with an identity, in which every nonzero element in $\mathbb{A}$ is a $unit$; that is, for ech $a\in \mathbb{A}$ with $a\ne 0$,there exists a unique element $a^{-1}$ such $a^{-1}\cdot a=a\cdot a^{-1}=1$.

Examples

The easiest examples of $rings$ are the traditional number systems. The set $\mathbb{Z}$ of the integers, with conventional additon and addition and multiplication, is a ring called the $ringoftheintegers$. We designate this ring simply with the letter $\mathbb{Z}$. Similarly, $\mathbb{Q}$ is the ring of the rational numbers, $\mathbb{R}$ is the ring of the real numbers; and $\mathbb{C}$ is the ring of the complex numbers. In each case, the operations are conventional addition and multiplication.

We must remember, that the elements of a ring are not necessarily numbers(for example, there are rings of polynomials/rings of functions, rings of switching circuits, and so on);and therefore "addition" does not necessarily refer to the conventional addition of numbers, nor does multiplications necessarily refer to the conventional operation of multiplying numbers. In fact, $+$ and $\cdot$ are nothing more than symbols denoting the two operations of a $ring$ .

for instance, $\mathcal{F}(\mathbb{R})$ represents the set of all the functions from $\mathbb{R}$ to $\mathbb{R}$;that is, the set fo all real-valued functions of a real variable:if $f$ and $g$ are any two functions from $\mathbb{R}$ to $\mathbb{R}$,their sum $f+g$ and their $productf\cdot g$ are defined as follows: $$[f+g](x)=f(x)+g(x)\text{for every real number x}$$ and $$[f\cdot g](x)=f(x)\cdot g(x)\text{for every real number x}$$ $\mathcal{F}(\mathbb{R})$ with these operations for adding and multiplying functions is a ring called $ringofrealfunctions$.It is written simply as $\mathcal{F}(\mathbb{R})$.

The rings $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$ and $\mathcal{F}(\mathbb{R})$ are all $infiniterings$, that is, rings with infinitely many elements. There are also $finiterings$: rings with a finite number of elements.As an important example, consider the group ${\mathbb{Z}}_n$,and define an operation of multiplication on ${\mathbb{Z}}_n$ by allowing the product $a\cdot b$ to be the remainder of the usual product of integers $a$ and $b$ after division by $n$(For example, in ${\mathbb{Z}}_5$， $2\cdot 4=3,3\cdot 3=4$, and $4\cdot 3=2$.)This operation is called $multiplicationmodulon$. ${\mathbb{Z}}_n$ with addition and multiplication modulo $n$ is a ring.