2.2: Rings
In this section, we'll seek to answer the questions:
- What are rings and integral domains, and how do they relate to fields?
- What are subrings, and how can we tell if a given subset of a ring is a subring?
- What special types of elements do rings have?
In the previous section, we observed that many familiar number systems are fields but that some are not. As we will see, these non-fields are often more structurally interesting, at least from the perspective of factorization; thus, in this section, we explore them in more detail. Before we proceed with that endeavor we will give a formal definition of polynomial so that we can include it in our work.
Let \(A\) be a set with a well-defined addition operation \(+\) and additive identity \(0\text{,}\) and \(x\) a variable. We define a polynomial in \(x\) with coefficients in \(A\) to be an expression of the form
\begin{equation*} p = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n\text{,} \end{equation*}
where \(a_n\ne 0\text{.}\) We call \(n\in \mathbb{N}_0\) the degree of the polynomial \(p\text{,}\) denoted \(\deg(p) = n\text{,}\) and \(a_0,a_1,\ldots, a_n\) the coefficients of the polynomial. The coefficient \(a_n\) is known as the leading coefficient of \(p\text{,}\) and \(a_n x^n\) is the leading term of \(p\text{.}\) By
\begin{equation*} A[x] := \{a_0 + a_1 x + a_2 x + \cdots + a_n x^n: n\in \mathbb{N}_0, \ a_i\in A \} \end{equation*}
we denote the set of all polynomials with coefficients in \(A\text{.}\) The additive identity of \(A[x]\) is \(0\text{,}\) called the zero polynomial , and is the polynomial whose coefficients are all \(0\text{.}\) The degree of the zero polynomial is \(-\infty\text{.}\)
Give some examples of polynomials in \(A[x]\) for various choices of number systems \(A\text{.}\) Identify their coefficients, leading terms, and degrees.
In the following table, fill in a Y if the set has the property; fill in a N if it does not.
| \(\mathbb{N}\) | \(\mathbb{Z}\) | \(2\mathbb{Z}\) | \(\mathbb{Q}\) | \(\mathbb{Q}[x]\) | \(\mathbb{Z}_{8}\) | \(\mathbb{Z}_{2}\) | \(\mathbb{R}\) | \(\mathbb{C}\) | \(\mathcal{M}_2(\mathbb{R})\) | |
|---|---|---|---|---|---|---|---|---|---|---|
| Closure under + | ||||||||||
| Closure under \(\cdot\) | ||||||||||
| \(+\) is associative | ||||||||||
| \(\cdot\) is associative | ||||||||||
| + is commutative | ||||||||||
| \(\cdot\) is commutative | ||||||||||
| \(\cdot\) distributes over + | ||||||||||
| There is an additive identity | ||||||||||
| All elements have additive inverses | ||||||||||
| There is an multiplicative identity | ||||||||||
| All nonzero elements have mult. inverses |
Which of the field axioms in Definition: Field hold for \(F[x]\text{,}\) where \(F\) is a field, and which fail to hold in general?
As a result of the answer to Exploration 2.2.3 and the completed Table 2.2.1 , we make the following definition.
A ring \(R\) is a nonempty set, together with binary operations \(+\) and \(\cdot\text{,}\) denoted \((R,+,\cdot)\text{,}\) and satisfying the following axioms.
- Given any \(a,b,c\in R\text{,}\) \((a+b)+c = a+(b+c)\text{.}\) (Associativity of addition)
- Given any \(a,b\in R\text{,}\) \(a+b= b+a\text{.}\) (Commutativity of addition)
- There exists an element \(0_R\in R\) such that for all \(a\in R\text{,}\) \(a+0_R = 0_R + a = a\text{.}\) (Additive identity)
- Given any \(a\in R\) there exists a \(b\in R\) such that \(a+b = b + a =0_R\text{.}\) (Additive inverses)
- Given any \(a,b,c\in R\text{,}\) \((a\cdot b)\cdot c = a\cdot (b\cdot c)\text{.}\) (Associativity of multiplication)
- For all \(a,b,c\in R\text{,}\) \(a\cdot (b+c) = a\cdot b + a\cdot c\text{.}\) (Distributive property I)
- For all \(a,b,c\in R\text{,}\) \((a+b)\cdot c = a\cdot c + b\cdot c\text{.}\) (Distributive property II)
As with fields, when the ring \(R\) is clear from context, we will often write \(0\) in place of \(0_R\text{.}\)
Compare and contrast Definitions: Field and Definition: Ring. What are the similarities? What are the differences?
While rings do not enjoy all the properties of fields, they are incredibly useful even in applied mathematics (see, e.g., Reference [1] for one recent example).
A ring \(R\) is said to be commutative if, for all \(a,b\in R\text{,}\) \(ab = ba\text{.}\) Additionally, \(R\) is said to have a unity or multiplicative identity if there is an element \(1_R\in R\) such that for all \(a\in R\text{,}\) \(a \cdot 1_R = 1_R \cdot a= a\text{.}\)
If \(R\) is noncommutative, it may have a left (respectively, right) identity, i.e., an element \(e\in R\) such that for all \(r\in R\text{,}\) \(er = r\) (respectively, \(re = r\)). If \(R\) has an element \(e\) for which \(er = re = r\) for all \(r\in R\text{,}\) \(e\) is often called a two-sided identity. In short, noncommutative rings may have left, right, or two-sided identities (or none at all).
Consider the sets given in Table 2.2.1 . Which are rings? Which are commutative rings with identity?
Which properties of fields in Theorem 2.1.1 hold for (commutative) rings?
Are all rings fields? Are all fields rings? Justify.
Most familiar rings are commutative, though not all. Most familiar (commutative) rings have identities, but not all. Find:
- A ring that does not have an identity 1 .
- A noncommutative ring that does have an (two-sided) identity.
Solution
- 1
-
Sometimes called a rng . \(\ddot\mathbb{S}mile\)
In the 1920s, Emmy Noether was the first to explicitly describe the ring axioms as we know them today, and her definition of a (not-necessarily-commutative) ring has led to a great deal of interesting work in algebra, number theory, and geometry, including the (see Section 3.3 for more on the historical development of the proof of Fermat's Last Theorem). Most modern definitions of ring agree with our Definition: Ring and allow for rings with noncommutative multiplication and no multiplicative identity.
The following theorem states that the set of polynomials with coefficients in a ring \(R\) is itself a ring under the usual operations of polynomial addition of like terms, and multiplication via distribution. The proof is not tricky, but a rigorous justification (especially of, e.g., the associativity of polynomial multiplication) is tedious, and thus is omitted.
If \(R\) is a (commutative) ring (with identity \(1_R\)), then \(R[x]\) is a (commutative) ring (with identity \(1_{R[x]} = 1_R\)).
One of the ways to better understand mathematical structures is to understand their similar substructures (e.g., given a vector space \(V\subseteq \mathbb{R}^n\) and a subspace \(W\subseteq V\text{,}\) we may write \(V = W + W^\perp\)).
Let \((R,+,\cdot)\) be a ring and let \(S\subseteq R\text{.}\) If \(S\) is itself a ring under \(+\) and \(\cdot\text{,}\) we say \(S\) is a subring of \(R\text{.}\) In this case, \(R\) is often called an overring of \(S\text{.}\)
The following theorem provides a easy-to-apply test to check if a given subset \(S\) of a ring \(R\) is in fact a subring of \(R\text{.}\)
Let \(R\) be a ring and \(S\) a subset of \(R\text{.}\) Then \(S\) is a subring if and only if:
- \(S\ne \emptyset\text{;}\)
- \(S\) is closed under multiplication; and
- \(S\) is closed under subtraction.
Determine whether the following rings \(S\) are subrings of the given rings \(R\text{.}\)
- \(S = \mathbb{Z}\text{,}\) \(R = \mathbb{Q}\)
- \(S = \mathbb{Z}_{5}\text{,}\) \(R = \mathbb{Z}_{7}\)
- \(S\) is any ring, \(R = S[x]\)
- \(S = \mathbb{R}\text{,}\) \(R = \mathbb{C}\)
In our study of rings, we are primarily interested in special types of subrings known as ideals , to be studied in more depth in Chapter 4.
Let \(R\) be a ring and let \(u\in R\) be nonzero. If there is a \(v\in R\) such that \(uv = vu = 1\text{,}\) we say \(u\) is unit of \(R\text{.}\) We denote the set of units of \(R\) by \(R^\times\text{.}\) We say \(x,y\in R\) are associates if there exists some \(u\in R^\times\) such that \(x = uy\text{.}\)
Explicitly describe the set \(\mathbb{Z}^\times\text{.}\) What are the associates of 7 in \(\mathbb{Z}\text{?}\)
In other words, a unit in a ring is a nonzero element with a multiplicative inverse. The existence of units is the primary difference between fields and commutative rings with identity: in a field, all nonzero elements are units, while in a commutative ring with identity, no nonzero elements need be units, as Theorem 2.2.2 demonstrates.
A commutative ring with identity \(R\) in which every nonzero element is a unit is a field.
A useful tool for analyzing the structure of rings with finitely many elements are addition and multiplication tables. As an example, consider the addition and multiplication tables for \(R = \mathbb{Z}_3\) shown in Table 2.2.2 and Table 2.2.3 .
| \(+\) | \(\overline{0}\) | \(\overline{1}\) | \(\overline{2}\) |
|---|---|---|---|
| \(\overline{0}\) | \(\overline{0}\) | \(\overline{1}\) | \(\overline{2}\) |
| \(\overline{1}\) | \(\overline{1}\) | \(\overline{2}\) | \(\overline{0}\) |
| \(\overline{2}\) | \(\overline{2}\) | \(\overline{0}\) | \(\overline{1}\) |
| \(\cdot\) | \(\overline{0}\) | \(\overline{1}\) | \(\overline{2}\) |
|---|---|---|---|
| \(\overline{0}\) | \(\overline{0}\) | \(\overline{0}\) | \(\overline{0}\) |
| \(\overline{1}\) | \(\overline{0}\) | \(\overline{1}\) | \(\overline{2}\) |
| \(\overline{2}\) | \(\overline{0}\) | \(\overline{2}\) | \(\overline{1}\) |
Calculate addition and multiplication tables for the following rings.
- \(\displaystyle R = \mathbb{Z}_5\)
- \(\displaystyle R = \mathbb{Z}_6\)
List 2-3 observations about your tables.
One of the interesting side effects of our definition of ring is that it allows for behavior that may at first appear unintuitive or downright weird.
A zero divisor in a ring \(R\) is a nonzero element \(z\in R\) such that there is a nonzero \(x\in R\) with \(zx = 0\) or \(xz=0\text{.}\)
Notice that the reason the idea of zero divisors at first appears weird is that they are not something we encounter when working with our familiar sets of numbers, such as \(\mathbb{Z}\) or \(\mathbb{R}\text{.}\) In fact, we specifically use the fact that there are no zero divisors in our familiar numbers systems to solve equations in high school algebra (e.g., if \((x-2)(x+5)=0\text{,}\) then \(x-2=0\) or \(x+5=0\)). The lack of zero divisors is one of the properties that does not persist in our abstraction from the integers to rings in general.
Find, with justification, all of the zero divisors in \(\mathbb{Z}_{10}\) and \(\mathbb{Z}_{11}\text{.}\) Make and prove a conjecture about the existence of zero divisors in \(\mathbb{Z}_m\text{,}\) where \(m > 1\text{.}\)
Are there any other rings in which you've seen zero divisors? Recall your answers to Exploration 2.2.4 .
Let \(R\) be a ring and suppose \(a,b\in R\) such that \(ab\) is a zero divisor. Then either \(a\) or \(b\) is a zero divisor.
Let \(R\) be a ring and \(u\in R^\times\text{.}\) Then \(u\) is not a zero divisor.
How can we reinterpret Investigation 1.4.1 in light of our new language of units and zero divisors? State a theorem that uses this new language.
While there is a well-developed body of literature on (noncommutative) rings (possibly without identity), from this point on, and unless stated otherwise, when we use the word ring we mean commutative ring with identity .
Moreover, while even commutative rings with identity and zero divisors are of interest to mathematicians, we will focus our study on rings with no zero divisors. As these rings share many properties of the integers, they are known as in tegral domains .
A commutative ring with identity \(R\) is an integral domain , or just domain , if \(R\) has no zero divisors.
The next activities and theorems help us identify examples of domains, as well as situate the notion of a domain in its proper place relative to fields and rings in general.
Which of the following rings are domains? Justify your answers.
- \(\displaystyle \mathbb{Z}\)
- \(\displaystyle \mathbb{Z}_{8}\)
- \(\displaystyle \mathbb{Z}_{19}\)
- \(\displaystyle \mathbb{R}\)
- \(\displaystyle \mathbb{Q}[x]\)
Every field is a domain.
Let \(m > 1\) and \(R = \mathbb{Z}_m\text{.}\) Then \(R\) is a field if and only if \(R\) is a domain.
If \(R\) is a domain and \(S\) is a subring of \(R\) with identity \(1_S = 1_R\text{,}\) then \(S\) is a domain.
If \(R\) is a domain, then so is \(R[x]\text{.}\)
Is the converse of Theorem 2.2.7 true? If so, give a short proof. If not, find a counterexample.
Given a field \(F\text{,}\) the set of polynomials \(F[x]\) is a domain.
When considering sets of polynomials, as we do in Chapter 3 (particularly in Section 3.1), the following results will be quite useful.
Let \(R\) be a domain, and let \(p(x),q(x)\in R[x]\) be nonzero polynomials. Then \(\deg(p(x) q(x)) = \deg(p(x)) + \deg(q(x))\text{.}\)
Can the hypotheses of Theorem 2.2.8 be relaxed? If so, provide more general hypotheses and adapt the proof. If not, give an illustrative example.
Let \(R\) be a domain. What are the units of \(R[x]\text{?}\) Prove your answer.