Definition of Rings

Rings

We spend the first chapter recalling the properties of $\mathbb{Z}$, the set of integers. The second chapter is dedicated to the study of ${\mathbb{Z}}_n$, the set of integers modulo $n$. We see both sets have two binary operations: addition and multiplication, and they satisfy the associative, commutative, and distributive properties. In this chapter, we will study the abstraction of these properties, the concept of rings.

Definition:

A ring is a set $R$ equipped with two binary operations $+$ and $\cdot$. The addition operation $+$ satisfies the following properties:

1. Closure: For all $a,b\in R$, $a+b\in R$.
2. Associative: For all $a,b,c\in R$, $(a+b)+c=a+(b+c)$.
3. Commutative: For all $a,b\in R$, $a+b=b+a$.
4. Additive Identity: There exists an element $0_R\in R$ such that for all $a\in R$, $a+0_R=a$.
5. Inverse: For all $a\in R$, there exists an element $-a\in R$ such that $a+(-a)=0_R$.

The multiplication operation $\cdot$ satisfies the following properties:

1. Closure: For all $a,b\in R$, $a\cdot b\in R$.
2. Associative: For all $a,b,c\in R$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.

The two operations satisfy the distributive property:

1. For all $a,b,c\in R$, $a\cdot (b+c)=a\cdot b+a\cdot c$ and $(a+b)\cdot c=a\cdot c+b\cdot c$.

Note we have put less requirements on the multiplication operation. We do not require multiplication to be commutative, nor do we require the existence of a multiplicative identity, nor do we require the existence of multiplicative inverses. When any of these happens, we will give the ring a special name, or at least a special adjective.

Extra Notations

We have the exponentiation notation for $a,b\in \mathbb{Z}$ with $b>0$:

$$a^b=\underset{b\text{ times}}{\underbrace{a\cdot a\cdots a}}.$$

We would like to generalize this notation to rings. For $a\in R$ and $n\in \mathbb{Z}$, we define:

Definition:

For $a\in R$ and $n\in {\mathbb{Z}}_{+}$, we define $a^n$ as:

$$a^n=\underset{n\text{ times}}{\underbrace{a\cdot a\cdots a}}.$$

If the ring $R$ has a multiplicative identity, we define $a^0=1_R$ for all $a\in R$.

Like the usual exponentiation, we have the following properties:

$$a^m\cdot a^n=a^{m+n}\text{ and }(a^m{)}^n=a^{m\cdot n}.\text{ for }m,n\in {\mathbb{Z}}_{+}.$$

We do not demand multiplication in rings to be commutative, so we do not have the property.

Warning:

The following property is not true in general:

$$(ab{)}^n=a^n{b}^n\text{ for }a,b\in R\text{ and }n\in {\mathbb{Z}}_{+}.$$

Remember here $n$ is a positive integer, there is generally no definition for exponent also an element of $R$. Namely, the following expression is not defined in general:

$$a^b\text{ for }a,b\in R.$$

We can also define the “multiplication” of $n$ by $a$ with $n\in {\mathbb{Z}}_{\ge 0}$ and $a\in R$:

Definition:

For $n\in {\mathbb{Z}}_{\ge 0}$ and $a\in R$, we define $na$ as:

$$na=\underset{n\text{ times}}{\underbrace{a+a+\cdots +a}}.$$

We define $0a=0_R$ for all $a\in R$.

Warning:

The above definition $0a=0_R$ is a definition. It does not prove us the following property:

$$0_R\cdot a=0_R\text{ for all }a\in R.$$

Though this is correct property which solely depends on the definition of ring, we defer the proof of this property to the next section.

Special Rings

Commutative Rings

Definition:

A ring $R$ is called a commutative ring if the multiplication operation is commutative. That is, for all $a,b\in R$, $a\cdot b=b\cdot a$.

The rings $\mathbb{Z}$ and ${\mathbb{Z}}_n$ are both commutative rings

There are many examples of non-commutative rings. The easiest one is $M_2(\mathbb{Z})$, the set of $2\times 2$ matrices with integer entries, with multiplication defined as matrix multiplication:

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{cc}e& f\\ g& h\end{array}\right)=\left(\begin{array}{cc}ae+bg& af+bh\\ ce+dg& cf+dh\end{array}\right)$$

Addition is defined as component-wise addition:

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)+\left(\begin{array}{cc}e& f\\ g& h\end{array}\right)=\left(\begin{array}{cc}a+e& b+f\\ c+g& d+h\end{array}\right)$$

One can easily find two matrices:

$$A=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)\text{ and }B=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$$

such that $A\cdot B\ne B\cdot A$.

Rings with Identity

Definition:

A ring $R$ is called a ring with identity if there exists an element $1_R\in R$ such that for all $a\in R$, $a\cdot 1_R=a$.

The rings $\mathbb{Z}$ and ${\mathbb{Z}}_n$ are both commutative rings with identity. The ring $M_2(\mathbb{Z})$ is a non-commutative ring with identity, the identity element being the $2\times 2$ identity matrix.

There are rings without identity. The easiest example is the set of even integers $2\mathbb{Z}$, which is a ring under the usual addition and multiplication operations. The additive identity is $0$, but there is no multiplicative identity.

Integral Domains

Definition:

A ring $R$ is called an integral domain if it is a commutative ring with identity, and has no zero divisors. That is, for all $a,b\in R$, if $a\cdot b=0_R$, then $a=0_R$ or $b=0_R$.

Warning:

I made a mistake in Feb. 12 lecture that I said the definition of integral domain does not require the ring to be commutative or have multiplicative identity. This is incorrect for this class. There are many variant of the definition of integral domain, for example Wikipedia page does not require multiplicative identity. We are taking the most restrictive definition of integral domain in this class. We have:

$$\text{Integral Domain}⊊\text{Commutative Ring}\cap \text{Ring with Identity}.$$

By the third property of ${\mathbb{Z}}_p$, we know ${\mathbb{Z}}_p$ is an integral domain if and only if $p$ is a prime number.

We can make the following definition, which can let us state the definition of integral domain more concisely.

Definition:

An element $a\in R$ is called a zero divisor if itself is not zero and there exists a non-zero element $b\in R$ such that $a\cdot b=0_R$.

Then we can restate the definition of integral domain as:

Definition:

A ring $R$ is called an integral domain if it is a commutative ring with identity, and has no zero divisors.

Fields

Definition:

A ring $R$ is called a field if it is a commutative ring with identity, and every non-zero element has a multiplicative inverse. That is, for all $a\in R$, $a\ne 0$, there exists an element $a^{-1}\in R$ such that $a\cdot a^{-1}=1_R$.

By the second property of ${\mathbb{Z}}_p$, we know ${\mathbb{Z}}_p$ is a field if and only if $p$ is a prime number.

Being a field is a stronger condition than being an integral domain. We state this as a proposition.

Proposition:

Every field is an integral domain.

Proof of Proposition

Let $F$ be a field, and assume $a,b\in F$ such that $a\cdot b=0$. If $a=0$, then we are done. If $a\ne 0$, then $a$ has a multiplicative inverse $a^{-1}$, so we have:

$$a\cdot b=0\Rightarrow a^{-1}\cdot a\cdot b=a^{-1}\cdot 0.$$

Now we can use this property we will shown next section. We can derive from ring axioms:

$$a^{-1}\cdot 0=0$$

Combined with:

$$a^{-1}\cdot a\cdot b=b$$

we have:

$$b=0.$$

$■$

Subrings

Definition:

A subset $S$ of a ring $R$ is called a subring if $S$ is a ring under the same operations of $R$. That is, $S$ is closed under addition and multiplication, and contains the additive identity and the additive inverses.

The set of even integers $2\mathbb{Z}$ is a subring of $\mathbb{Z}$, but the set of odd integers is not a subring. The set of $2\times 2$ matrices with even entries is a subring of ${\mathbb{M}}_2(\mathbb{Z})$. Here is an interesting subring of $\mathbb{Q}$:

Example:

S is the set of all rational numbers $a/b$, where when $a/b$ is written in reduced form, $b$ is not divisible by 3. We can check $S$ is a subring of $\mathbb{Q}$.

Build New Rings

Cartesian Product

Finding subrings in an existing ring is a common way to build new rings. We can also build new rings by taking the Cartesian product of two rings.

Definition:

Let $R$ and $S$ be rings. The Cartesian product $R\times S$ is a ring under the operations:

$$(a,b)+(c,d)=(a+c,b+d)\text{ and }(a,b)\cdot (c,d)=(a\cdot c,b\cdot d).$$

The Cartesian product of two rings is a ring. The two factors are subrings of the product ring.

Exercise:

The subsets $R\times \{0\}$ and $\{0\}\times S$ are subrings of $R\times S$.

Matrix Rings

We can also build new rings by taking the set of $n\times n$ matrices with entries in a ring $R$. When do this type of construction, we usually require the ring $R$ to be commutative.

Definition:

Let $R$ be a commutative ring. The set $M_n(R)$ of $n\times n$ matrices with entries in $R$ is a ring under the usual matrix addition and multiplication operations.

Note that the ring $M_n(R)$ is not in general commutative when $n>1$. The matrix construction is a way to build non-commutative rings.

Polynomial Rings

Another way to build new rings is to take the set of polynomials with coefficients in a ring. Here is the definition:

Definition:

Let $R$ be a commutative ring. The set $R[x]$ of polynomials with coefficients in $R$ is a ring under the usual polynomial addition and multiplication operations.

Note since $R$ is a commutative ring, then $R[x]$ is also a commutative ring. The polynomial ring is a way to build new commutative rings.