7.3 Rings and Fields

In this section we are going to first explore several different examples of sets that have a standard pair of two different binary operations, addition and multiplication. We will create categorizations of these sets based upon their properties. Some properties to consider are associative, commutative, distributive, identities, and inverses.

Example 7.8 Consider the following sets and operations.

\(\mathbb{N}\) with the usual addition and multiplication.
\(\mathbb{Z}\) with the usual addition and multiplication.
\(\mathbb{Q}\) with the usual addition and multiplication.
\(\mathbb{R}\) with the usual addition and multiplication.
\(\mathbb{C}\) with the usual addition and multiplication.
\(2\mathbb{Z}\) (the even integers) with the usual addition and multiplication.

For each of these sets with operations, determine which of the following properties hold.

Additive identity
Additive inverse
Additive commutativity
Multiplicative identity
Multiplicative inverse
Multiplicative commutativity
Distribution properties of multiplication over addition

7.3.1 Finite Sets with Two Operations

While each of the sets listed previously have infinite cardinality, a large portion of abstract algebra studies finite sets, combined with appropriate operations. \(\mathbb{Z}_n = \left\{0, 1, 2, \ldots, n-1\right\}\) with addition and multiplication defined by the remainder of the sum and product when divided by \(n\). We provide some examples of these types of sets and operations below. For these finite sets, it is often easiest to just provide a table of the operations, rather than an algebraic definition.

Example 7.9 Let \(\mathbb{Z}_2=\left\{ 0,1 \right\}\) with the operations given in the following Cayley tables. \[\begin{array}{cc} \begin{array}{c|cc} + & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} & \begin{array}{c|cc} \cdot & 0 & 1 \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \end{array}\]

Example 7.10 Let \(\mathbb{Z}_3=\left\{ 0,1, 2 \right\}\) with the operations given in the following Cayley tables. \[\begin{array}{cc} \begin{array}{c|ccc} + & 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 2\\ 1 & 1 & 2 & 0 \\ 2 & 2 & 0 & 1 \\ \end{array} & \begin{array}{c|ccc} \cdot & 0 & 1 & 2\\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 1 \end{array} \\ \end{array}\]

Example 7.11 Let \(\mathbb{Z}_4=\left\{ 0,1, 2, 3 \right\}\) with the operations given in the following Cayley tables. \[\begin{array}{cc} \begin{array}{c|cccc} + & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 1 & 2 & 3 \\ 1 & 1 & 2 & 3 & 0 \\ 2 & 2 & 3 & 0 & 1 \\ 3 & 3 & 0 & 1 & 2 \\ \end{array} & \begin{array}{c|cccc} \cdot & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 2 & 0 & 2 \\ 3 & 0 & 3 & 2 & 1 \\ \end{array}\\ \end{array}\]

Example 7.12 Let \(\mathbb{Z}_6=\left\{ 0,1, 2, 3, 4, 5 \right\}\) with the operations given in the following Cayley tables. \[\begin{array}{cc} \begin{array}{c|cccccc} + & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 \\ 1 & 1 & 2 & 3 & 4 & 5 & 0\\ 2 & 2 & 3 & 4 & 5 & 0 & 1 \\ 3 & 3 & 4 & 5 & 0 & 1 & 2 \\ 4 & 4 & 5 & 0 & 1 & 2 & 3 \\ 5 & 5 & 0 & 1 & 2 & 3 & 4 \\ \end{array} & \begin{array}{c|cccccc} \cdot & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5\\ 2 & 0 & 2 & 4 & 0 & 2 & 4 \\ 3 & 0 & 3 & 0 & 3 & 0 & 3 \\ 4 & 0 & 4 & 2 & 0 & 4 & 2 \\ 5 & 0 & 5 & 4 & 3 & 2 & 1 \\ \end{array} \end{array}\]

These different sets (and operations) satisfy various properties depending upon the number of elements in the set. It is a valuable exercise to explore how the these are connected.

7.3.2 Matrices

We let \(GL(n,\mathbb{R})\) be the set of \(n\times n\) matrices with coefficients from the real numbers with the usual matrix addition and multiplication. We can see that this set has an additive identity, \(0\), and multiplicative identity, \(I\). \[0 = \begin{pmatrix} 0 & 0 & 0 & \cdots & 0 \\ 0 & 0 & 0 & \cdots & 0 \\ 0 & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 0 \end{pmatrix} \quad \mbox{ and } \quad I= \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix}\]

It also satisfies most of the properties of our number systems that we have explored, including the commutative property of addition. However, we can discover that multiplication is not commutative, not every element has a multiplicative inverse, and we can have the product of two non-zero elements be zero. For example, if we look at \(GL(2,\mathbb{R})\) and the matrices \[A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \quad \mbox{ and } \quad B= \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.\]

Then \(A^2=0\), \(B^2=0\), \[AB=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \quad \mbox{ and } \quad BA= \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.\]

Sometimes we combine ideas of the finite sets with the additional structure of matrices. For example, we let \(GL(2,\mathbb{Z}_2)\) be the two by two matrices with coefficients from \(\mathbb{Z}_2\), with the usual operations of matrix addition and multiplication.

Related Content Standards

(HSN.VM.8) Add, subtract, and multiply matrices of appropriate dimensions.
(HSN.VM.9) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
(HSN.VM.10) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

7.3.3 Rings

Now that we have several examples of sets with two binary operations defined on them, we see that there are certain properties of these binary operations that are very useful for understanding the structure of the set.

Definition 7.5 A ring \(\langle R,+,\cdot \rangle\) is a set \(R\) together with two binary operations \(+\) and \(\cdot\), which we call addition and multiplication, defined on \(R\) such that the following are satisfied:

\(\left( R,+\right)\) is an abelian group.
- G1: The binary operation \(+\) is associative.
- G2: There is an element \(0\) in \(R\) such that \(0+a=a+0=a\) for all \(a\in R\). (This element is called the additive identity of \(R\).)
- G3: For each \(a\) in \(R\) there is an element \(-a\) in \(R\) such that \[(-a)+a=a+(-a)=0.\] (This element is called the additive inverse of \(a\) in \(R\).)
- For each \(a\) and \(b\) in \(R\), \(a+b=b+a\). (This means that the operation is commutative and so the group is abelian.)
The binary operation \(\cdot\) is associative.
For all \(a,b,c\in R\), the left distribution law, \(a(b+c)=(ab)+(ac)\), and the right distribution law, \((a+b)c=(ac)+(bc)\), hold.

We can see from this definition that the integers, \(\langle\mathbb{Z},+,\cdot\rangle\), form a ring but the natural numbers, \(\langle\mathbb{N},+,\cdot\rangle\), do not because they do not have additive inverses. Similarly, during the constructions of the rational numbers, \(\langle\mathbb{Q},+,\cdot\rangle\); the real numbers, \(\langle\mathbb{R},+,\cdot\rangle\); and the complex numbers, \(\langle\mathbb{C},+,\cdot\rangle\), we proved that these number systems satisfied all of the ring properties.

We can also see that the examples above of \(\langle\mathbb{Z}_n,+,\cdot\rangle\) and \(\langle GL(n,\mathbb{R}),+,\cdot\rangle\) are also rings. So in many ways, all of these sets and operations have some very similar properties. However, there are some properties that some of these rings have, that others do not. If a ring has certain additional properties, we will add some labels to these sets.

Definition 7.6 (Further definitions) We classify rings based on various properties of the ring.

A ring in which the multiplication is commutative is called a commutative ring.
A ring with a multiplicative identity is called a ring with unity.
If every non-zero element of a ring has a multiplicative inverse, then the ring is called a division ring.
A field is a commutative division ring.

Related Content Standards

(7.NS2) Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
1. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as \((-1)(-1) = 1\) and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

7.3.4 Identities and Inverses

Now that we have created several categories and labels for the sets and operations, let’s explore some of the consequences of the properties on the operations. As we do so, keep in mind which properties are used to prove different results. That way you can determine which properties are required of various sets and operations in order to get those results later on.

Theorem 7.8 The additive identity for a ring is unique.

This theorem is a direct consequence of the set with addition forming a group, whose identity is unique. Similarly, we have that the multiplicative identity is unique (if it exists). Since the proof is so similar to that of the uniqueness of the additive identity, the proof is left to the reader as an exercise.

Theorem 7.9 If a ring has a multiplicative identity, then that multiplicative identity is unique.

Now that we know that the additive and multiplicative identities of a ring are unique, we will usually label these identities as \(0\) and \(1\), respectively.

Our next theorem shows us that the property from the integers that multiplying any number by zero results in zero is not unique to our number systems, but is inherent in the properties of the additive identity in a ring.

Theorem 7.10 If \(\langle R,+,\cdot\rangle\) is a ring, and if \(0\) is the additive identity in \(R\), \(0\cdot a=a\cdot 0=0\) for all \(a\in R\).

Proof. Let \(\langle R,+,\cdot\rangle\) be a ring with additive identity \(0\) and let \(a\in R\). Then \(a \cdot 0 = a \cdot (0+0)\), since \(0\) is the additive identity. From the distribution property we have that \(a \cdot 0 = a \cdot 0 + a \cdot 0\) and if we add the additive inverse of \(a \cdot 0\) to both sides of the equation, we have that \(0=a\cdot 0\). We can similarly show that \(0\cdot a=0\).

Recall from Theorem 7.2 that we have uniqueness of the additive inverse since \(\langle R,+\rangle\) is an abelian group. We can also prove the uniqueness of a multiplicative inverse, if such an inverse exists because in this situation, \(\langle R,\cdot \rangle\) forms a group.

One question that students often ask when they are introduced to multiplication with negative numbers is why a negative times a negative is a positive. We see from the theorem below that this property, and other related properties are inherited from the ring structure of the number system. However, when explaining this property to students, it is often better to focus on the symmetry about the additive identity. In particular, with integers and real numbers it is good to focus on the symmetry of the additive identities on the number line. For the complex numbers it is helpful to focus the students’ attention on the symmetry about the additive identity when looking at the complex plane.

Theorem 7.11 If \(\langle R,+,\cdot\rangle\) is a ring and \(a,b \in R\), then

\(a\cdot (-b)=(-a)\cdot b=-(a\cdot b)\)
\((-a)\cdot (-b)=a\cdot b\)

Proof. In order to prove the first part of the theorem we need to only show that \[\left( a \cdot (-b) \right) + \left(a \cdot b\right) =0 \quad \mbox{and} \quad \left((-a)\cdot b\right)+\left(a \cdot b\right) =0\] because the additive inverse of \(a\cdot b\) is unique.

We see that \[\begin{align*} (a \cdot (-b)) + (a \cdot b ) &= a \cdot ((-b)+b) \mbox{ (from the distributive property)} \\ &= a \cdot 0 \mbox{ (since } -b \mbox{ is the additive inverse of }b)\\ &= 0 \end{align*}\] as a result of Theorem 7.10. Similarly, \[\begin{align*} ( (-a) \cdot b) + (a \cdot b) &= ((-a)+a)\cdot b \mbox{ (from the distributive property)} \\ &= 0 \cdot b \mbox{ (since } -a \mbox{ is the additive inverse of } a)\\ &= 0 \end{align*}\] as a result of Theorem 7.10. For the second result we use the first result to see that \[(-a)\cdot (-b) = -\left(a \cdot (-b)\right) = - \left( - \left(a\cdot b\right) \right) = a \cdot b\] since the additive inverse of the additive inverse of an element is the original element.

7.3.5 Integral Domains

We often want to solve an equation such as \(ax=b\) or \(xa=b\) for \(x\) with \(a\neq 0\). In order to do so we need to use the cancellation laws, \(ab=ac\) with \(a \neq 0\) implies \(b=c\) and \(ba=ca\) with \(a\neq 0\) implies \(b=c\).

We will see that the cancellation laws correspond with a property about zero divisors.

Definition 7.7 If \(a\) and \(b\) are two non-zero elements of a ring \(R\) such that \(ab=0\), then \(a\) and \(b\) are divisors of \(0\) (or zero divisors). In particular, \(a\) is a left divisor of \(0\) and \(b\) is a right divisor of \(0\).

Theorem 7.12 The cancellation laws hold in a ring \(R\) if and only if \(R\) has no left or right divisors of \(0\).

Proof. Assume that for all elements \(a\), \(b\), and \(c\) of a ring \(R\) such that \(a\neq 0\) we have that \[(ab=ac) \Rightarrow (b=c).\] We can then add the additive inverse of \(ac\) to both sides of the first equation and the additive inverse of \(c\) to both sides of the right equation to see that this implication is equivalent to \[\left((ab-ac)=0 \right) \Rightarrow (b-c=0).\] Using the left distribution property of the ring, we have that the implication is equivalent to \[\left( a \cdot (b-c) =0\right) \Rightarrow \left( (b-c)=0\right).\] This implication is true if and only if \(a\) is not a left divisor of \(0\).

Similarly, one can show that the right cancellation law of \[\left(ba=ca, \: \mbox{ with } a \neq 0 \right) \Rightarrow (b=c)\] is equivalent to \(a\) not being a right divisor of \(0\).

Therefore, we see that the cancellation laws holding for a ring is equivalent to the ring having no zero divisors.

Since this property of not having zero divisors is so important in the process of solving simple equations, we give rings with this property a label.

Definition 7.8 An integral domain is a commutative ring with unity containing no divisors of \(0\).

A very useful example of an integral domain is the set of integers with the usual addition and multiplication. We will see in the next chapter that the polynomials also form an integral domain, which allows us to factor polynomials in a similar way to factoring integers.

In order to better understand how all of these different types of rings fit together, Figure 7.1 is a graphical way to understand their relationships.

Figure 7.1: Venn diagram of ring relationships

7.3.6 Ring Homomorphisms

Just like group homorphisms in Section 7.1.3, we can study the functions between rings that maintain the ring structures.

Definition 7.9 Let \(\langle R,+,\cdot\rangle\) and \(\langle R',\ast,\times\rangle\) be rings. A map \(\phi:R\rightarrow R'\) is a ring homomorphism if

\(\phi(a+b)=\phi(a) \ast \phi(b)\) for all \(a\) and \(b\) in \(R\) (\(\phi\) maps addition to addition)
\(\phi(a\cdot b)=\phi(a)\times \phi(b)\) for all \(a\) and \(b\) in \(R\). (\(\phi\) maps multiplication to multiplication)

\(\phi\) is called a ring isomorphism if it is also a bijection. If such an isomorphism exists from a ring \(\langle R,+,\cdot\rangle\) to a ring \(\langle R',\ast,\times\rangle\), then we say that \(\langle R,+,\cdot\rangle\) is isomorphic to \(\langle R',\ast,\times\rangle\).

When we talk about rings, we often refer to a ring \(\langle R,+,\cdot\rangle\) as just \(R\) when the binary operations are easily inferred from the set. For instance, we often talk about \(\mathbb{R}\) as the ring of real numbers with the binary operations of addition and multiplication being inferred.

Similar to groups, it is useful to know when two rings are really the “same” ring with different labels. We see that just like with groups (Theorem 7.7), ring isomorphisms create an equivalence relation. So two rings are the “same” if there is an isomorphism from one to the other. Therefore, in order to prove that two rings are isomorphic, it is usually done by finding the isomorphism. However, to prove that two rings are not isomorphic we usually use a proof by contradiction using certain properties of ring isomorphisms.

7.3.7 Exercises

Create Cayley tables for addition and multiplication for \(\mathbb{Z}_{12}\). What patterns do you recognize inside of the table and how would you explain the reasons for those patterns?
What are the zero divisors of \(GL(2,\mathbb{Z}_2)\)?
For each of the following rings, determine the appropriate location for the ring in the Venn diagram of the ring relationships (Figure 7.1).
1. \(\langle\mathbb{Z},+,\cdot\rangle\), the integers with the usual addition and multiplication.
2. \(\langle\mathbb{Q},+,\cdot\rangle\), the rational numbers with the usual addition and multiplication.
3. \(\langle\mathbb{R},+,\cdot\rangle\), the real numbers with the usual addition and multiplication.
4. \(\langle\mathbb{C},+,\cdot\rangle\), the complex numbers with the usual addition and multiplication.
5. \(\langle2\mathbb{Z},+,\cdot\rangle\), the even integers with the usual addition and multiplication.
6. \(\langle\mathbb{Z}_2,+,\cdot\rangle\), as defined in Example 7.9.
7. \(\langle\mathbb{Z}_4,+,\cdot\rangle\), as defined in Example 7.11.
8. \(\langle GL(2,\mathbb{R}),+,\cdot\rangle\), the two by two matrices with real coefficients with the usual matrix addition and multiplication.
9. \(\langle GL(2,2\mathbb{Z}),+,\cdot\rangle\), the two by two matrices with even integer coefficients with the usual matrix addition and multiplication.
Prove Theorem 7.9.
Show that \(a^2-b^2=(a+b)(a-b)\) for all \(a\) and \(b\) in \(R\) if and only if R is a commutative ring.
Prove that \(2\mathbb{Z}\) and \(3\mathbb{Z}\) are not isomorphic as rings.
Let \(R\) be a ring and let \(R^R\) be the set of all functions mapping \(R\) into \(R\). For \(f,g\in R^R\), define the sum \(f+g\) by \[(f+g)(a)=f(a)+g(a)\] and the product \(f\cdot g\) by \[(f\cdot g)(a)=f(a)\cdot g(a)\] for each \(a \in R\). Show that \(\langle R^R, +, \cdot\rangle\) is a ring.
Vector spaces, or specifically Euclidean spaces, constitute a subject of significant study in secondary and tertiary mathematics.
1. Discuss how vector spaces are related to groups, rings, and fields by describing their similarities differences or inclusions.
2. In what ways are linear transformations between vector spaces related to homomorphisms of groups or rings.