# 6.1: Introduction to Rings

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Recall a group is a set with a binary operation; rings are algebraic structures similar to groups but with two operations instead of one.

##### Definition: Ring

A non-empty set $$R$$ with two binary operations, addition and multiplication - denoted by $$+$$ and $$\bullet$$, is called a ring if:

1. $$(R,+)$$ is an abelian group .

2. $$a(bc)=(ab)c, \; \forall a,b,c \in R$$.  $$R$$ in this context is a ring.  Note $$(R,\bullet)$$ is a semigroup.

3. $$a \bullet (b+c)=ab+ac$$ and $$(b+c)a=ba+ca$$, $$\forall a,b,c \in R$$.  Distributive property.

##### Definition: Commutative Ring

A ring $$(R, +, \bullet)$$ is called a commutative ring if $$ab=ba, \; \forall a,b \in R$$.

##### Definition: Unity

A ring $$(R, +, \bullet)$$ has a unity(identity) $$e \in R$$ if $$ae=ea=a, \; \forall a \in R$$.  Note unity is always a unique, single element.

##### Definition: Unit

An element $$a\in R$$ is called a unit if there exists an $$a^{-1}\in R$$ s.t. $$aa^{-1}=e$$.

##### Example $$\PageIndex{1}$$
1. $$(\mathbb{Z},+, \bullet)$$ is a commutative ring with unity $$1$$, and  each $$a \in \mathbb{Z} \setminus \{0\}$$ is a unit.
2. $$(\mathbb{Z}_n,\bigoplus, \bigodot)$$ (modulo n addition and multiplication) is a commutative ring with unity 1 and units of $$U(n)$$ since they must be all the elements relatively prime to n.

3. $$(\mathbb{Z}_{[x]}, +, \bullet)$$ where $$\mathbb{Z}_{[x]}$$ is the set of all integer polynomials in variable $$x$$, is a commutative ring with unity of 1 and units of $$\pm 1$$.

4. $$(M_{22}(\mathbb{Z}), +, \bullet)$$ is a ring, from the set of all $$2 \times 2$$ matrices with integer entries, that has unity $$I$$.  It is a non-commutative ring.  The units are the set of all $$2 \times 2$$ matrices with a non zero determinant  $$(GL_2(\mathbb{R}))$$.

5. $$(\mathbb{R},+, \bullet)$$ is called a field.  It is a commutative ring, and inverses exist for all elements except 0.

6. $$(\mathbb{C},+, \bullet)$$ is also a field, and inverses exist for all elements except 0.

7. A polynomial ring $$R[x]$$  over a ring $$R$$ is defined as $$\{(p(x)=a_0+a_1x+\cdots+a_nx^n| n \in \mathbb{Z}, n\geq 0, a_i \in R, \forall i=1, \cdots, n\}$$. Where $$a_i$$  is the coefficient of $$x^i$$, also called the coefficients of the polynomial, and the non-negative integer $$n$$ is called the degree of the polynomial.

The coefficients $$a_i$$ can be elements from any ring, such as integers, rational numbers, real numbers, complex numbers, or even other polynomials. The addition and multiplication  of $$R[x]$$ are the addition and multiplication $$R.$$

Integral Domain (ID)

Recall: $$(R,+,\bullet)$$ is a non-empty set with closed binary operations.

$$(R,+)$$ is abelian.

$$(R,\bullet)$$ is a semigroup.  Recall this is a set with an associative operation, no identity.

$$(R,+,\bullet)$$ is distributive, has unity (aka identity) of 1 and unit of $$aa^{-1}=e$$

If $$ab=ba$$, the ring is commutative on $$\bullet$$.

##### Definition:  Integral Domain

A commutative ring with unity (aka identity), $$R$$, is called an integral domain if for every $$a,b \in R$$, $$ab=0$$ implies $$a=0$$ or $$b=0$$.

##### Example $$\PageIndex{2}$$

An example of a non-integral domain is

$$\begin{bmatrix} 1 & 0 \\0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\0 & 1 \end{bmatrix}=\begin{bmatrix} 0 & 0 \\0 & 0 \end{bmatrix}$$, but neither $$\begin{bmatrix} 1 & 0 \\0 & 0 \end{bmatrix}$$ nor $$\begin{bmatrix} 0 & 0 \\0 & 1 \end{bmatrix}$$ equal zero.

##### Definition: Division Ring

A division ring is a ring, (R), with an identity in which for every non-zero $$a \in R$$, there exists $$b \in R$$ s.t. $$ab=1=ba$$.

##### Definition: Field

A field is a commutative division ring.

##### Example $$\PageIndex{3}$$

Examples of fields are:  $$(\mathbb{Q},+,\bullet)$$, $$(\mathbb{R},+,\bullet)$$, and $$(\mathbb{C},+,\bullet)$$, with the later being algebraically closed. ##### Definition: Zero divisor

A non-zero element of $$a \in R$$ is called a zero divisor if there exists $$ab \in R$$ s.t. $$ab=0$$.

##### Example $$\PageIndex{4}$$

$$(\mathbb{Z},+,\bullet)$$.

Since there are no zero divisors, it is an integral domain.   It is a commutative ring.  Since zero does not have an inverse, not every element has an inverse. Thus, it is not a field.

##### Example $$\PageIndex{5}$$

$$(\mathbb{Z}_n,\oplus,\odot)$$. There are two cases.

This is a finite field if $$n$$ is prime.

If $$n$$ is not prime, then it is not an integral domain since it will have a zero divisor.

##### Example $$\PageIndex{6}$$

The set of all real-valued differentiable functions on $$[a,b]$$ is a commutative ring with a unit.  This area of study is called differential algebra.

##### Example $$\PageIndex{7}$$

$$M_{22}(\mathbb{R},+,\bullet))$$.

The addition is abelian.  It is not commutative.  It has an identity. Not an integral domain.

This area of study is called matrix algebra.

##### Example $$\PageIndex{8}$$

Let

$$1=\begin{bmatrix} 1 & 0 \\0 & 1 \end{bmatrix}, I= \begin{bmatrix} 0 & 1 \\-1 & 0 \end{bmatrix}, J= \begin{bmatrix} 0 & i \\i & 0 \end{bmatrix}, K= \begin{bmatrix} i & 0 \\0 & -i \end{bmatrix}$$.

Define

$$R= \{ a+bI+cJ+dK |a,b,c,d \in \mathbb{R} \}.$$ Then $$R$$ is Ring of Quaternions.

##### Example $$\PageIndex{9}$$

Let

$$\mathbb{H}= \left \{ \begin{bmatrix} a & b \\- \bar{b} & \bar{a} \end{bmatrix} | a, b \in \mathbb{C} \right\}.$$

Then  $$\mathbb{H}$$ is a division ring with matrix addition and multiplication, but not an integral domain.

## Properties of Rings

##### Theorem $$\PageIndex{1}$$: Cancellation Law Let $$D$$ be a commutative ring with identity. Then $$D$$ is an integral domain if and only if $$ab=bc$$ implies $$b=c\; \forall a,b,c \in D$$.

##### Theorem $$\PageIndex{2}$$

Let $$(R,+, \bullet)$$ be a ring and $$a,b \in R$$.

1. $$a0=0=0a$$.

2. $$a(-b)=(-a)b=-ab$$.

3. $$(-a)(-b)=(ab)$$.

1. Let $$a \in R.$$ Then $$0 a=(0+0) a=0a+0a. \implies 0a+0=0a+0a.$$ By cancellation law of $$+$$, we get $$0a=0.$$ Similarly, we can show that $$a 0=0.$$

2. Let $$a, b \in R.$$ Then $$ab+(-a)b=(a+(-a))b=0 b=0. \implies (-a)b=-(ab).$$ Similarly, $$ab+a(-b)=a(b+(-b))=a 0=0. \implies a(-b)=-(ab).$$

## Subring:

Let $$(R,+,\bullet)$$ be a ring.

Let $$S(\ne 0) \subseteq R$$.

Then $$S$$ is a subring of $$R$$ if $$(S,+,\bullet)$$ is a ring with the same operations.

##### Theorem $$\PageIndex{3}$$

Subring Test

Let $$(R,+,\bullet)$$ be a ring.

$$S \subseteq R$$ is a subring if and only if the following conditions are satisfied:

1. $$S \ne \{\}$$.  At least $$0\in S$$.

2. $$a-b\in S$$.

3. $$ab \in S, \; \forall a,b \in A$$.

##### Example $$\PageIndex{10}$$

Let $$(R,+,\bullet)$$ be a ring then $$\{0\}$$ and $$R$$ are subrings of $$R$$ and are called the trivial subrings of $$R$$.

##### Example $$\PageIndex{11}$$

Let $$(\mathbb{Q},+, \bullet)$$ is a subring of $$(\mathbb{R},+, \bullet)$$ and $$(\mathbb{R},+,\bullet)$$ is a subring of $$(\mathbb{C},+,\bullet)$$.

##### Example $$\PageIndex{12}$$

$$(\mathbb{Z}_6,\oplus, \odot)$$ is a ring.

1.  Let $$S=\{0, 3 \}$$.

Clearly, $$S$$ is non-empty.

Since $$0\oplus 3$$, $$0 \oplus0$$, and $$3 \oplus3$$, $$a-b \in S$$.

Since $$0\odot3$$, $$0 \odot 0$$, and $$3 \odot 3$$, then $$ab \in S$$ .

Thus $$S$$ is a subring of $$(\mathbb{Z}_6,\oplus, \odot)$$.

2.

Let $$T =\{0, 2, 4 \}$$.

Clearly, $$T$$ is non-empty.

Since $$0 \oplus 0$$, $$0 \oplus 2$$, $$0 \oplus 4$$, $$2 \oplus 2$$,$$2 \oplus 4$$ and $$4 \oplus 4\in T$$ then $$a-b \in T$$

Since $$0 \odot 0$$,  $$0 \odot 2$$,  $$0 \odot 4$$,  $$2 \odot 2$$,  $$2 \odot 4$$, and  $$4 \odot 4 \in T$$, then $$ab \in T$$.

Thus $$T$$ is a subring of  $$(\mathbb{Z}_6,\oplus, \odot)$$.

##### Example $$\PageIndex{13}$$

Define $$\mathbb{Z}[i]=\{ m+ni|m.n \in \mathbb{Z}\},$$ where $$i^2=-1.$$

This is a ring called Gaussian integers. The units of this ring are $$\pm1$$ and $$\pm i$$. Hence  The ring of Gaussian integers is not a field.This is a subring of $$(\mathbb{C},+,\cdot ).$$

##### Theorem $$\PageIndex{4}$$: Wedderburn's Theorem

Every finite integral domain is a field.

Let $$D$$ be a finite integral domain.

Let $$D^{*} = D \backslash \{0\}$$.

We shall show that every element in $$D^*$$ is a unit (i.e., $$aa^{-1}=e,\; \forall a \in D^*$$).

Let $$a\in D^*$$.

Define $$\lambda_a : D^* \rightarrow D$$ by $$\lambda_a(x)=ax, \; x\in D^*$$.

We shall show that $$\lambda_a$$ is injective.

Let $$x_1, x_2 \in D^*$$ s.t. $$\lambda_a(x_1)=\lambda_a(x_2)$$.

Consider $$ax_1=ax_2$$.

Thus $$ax_1-ax_2=0$$.

So $$a(x_1-x_2)=0$$.

Since $$D$$ is an integral domain, either $$a=0$$ or $$(x_1-x_2)$$=0\).\)

Since $$D^*$$ excludes zero, $$a \ne 0$$. Thus $$x_1-x_2=0$$.

Hence $$x_1=x_2$$ and $$\lambda_a$$ is injective.

Since $$D$$ and $$D^*$$ are finite, $$\lambda_a$$ is surjective.

Since $$1 \in D^*, \; \exists x \in D^*$$ s.t. $$\lambda_a(x)=1$$.

Thus $$a$$ is a unit, and therefore it is a field

## Characteristic of a ring R

##### Definition: Characteristic

Let $$r \in R$$, then $$r+r=2r, r+r+r=3r, \cdots, r+ \cdots+ r =nr.$$

The characteristic of a ring $$R$$  is the least positive integer $$n$$ such that $$nr=0, \forall r \in R.$$ If no such $$n$$ exists, we say that the characteristic of ring $$R$$ is $$0.$$

##### Example $$\PageIndex{14}$$
1. $$( \mathbb{Z}, + \cdot), ( \mathbb{Q}, + \cdot), ( \mathbb{R}, + \cdot), ( \mathbb{C}, + \cdot)$$ are rings of characteristic $$0.$$

2. $$(\mathbb{Z}_6,\oplus, \odot)$$ is a ring of characteristic $$6.$$

##### Theorem $$\PageIndex{5}$$
1. Let $$R$$ be ring with identity $$1.$$ If $$n$$ is the least positive integer such that $$n1=0,$$ then the characteristic of ring R is $$0.$$

2. The characteristic of an integral domain is  either prime or $$0.$$

This page titled 6.1: Introduction to Rings is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.