Chapter 16 - Rings
Section 16.1 - Rings
Definition. A nonempty set \(S\) is a ring if, with two binary operations called addition and multiplication, the following are satisfied:
- Addition is commutative. \(a + b = b + a\) for \(a, b \in R\)
- Addition is associative. \((a + b) + c = a + (b + c)\) for \(a, b, c \in R\)
- There exists a zero-element \(0_R\) in \(R\) such that \(a + 0 = a\) for all $a \in $
- Every element \(a\) has an additive inverse \(-a \in R\) such that \(a + (-a) = 0_R\)
- Multiplication is associative. That is, \(a(bc) = (ab)c\) for \(a, b, c \in R\)
- The Distributive Property holds. That is, \(\forall a, b, c \in R,\)
Definition. If there exists some element \(1_R \in R\) such that \(1a = a1 = a\) for all \(a \in R\), we say that \(R\) is a ring with unity or identity.
Note that some books impose the condition that \(1 \neq 0\). If \(1 = 0\), we can show the ring only has one element.
Definition. If \(ab = ba\) for all \(a, b \in R\), the ring is said to be a commutative ring.
Definition. If a ring \(R\) is commutative, \(R\) is an integral domain if and only if for every \(a, b \in R\), \(ab = 0\) implies that either \(a = 0\) or \(b = 0\).
Definition. An element \(a \in R\) is called a unit if there exists some \(a^{-1}\) such that \(a a^{-1} = a^{-1} a = 1\).
Definition. A ring \(R\) with identity is called a division ring if every nonzero element in \(R\) is a unit.
Definition. A commutative division ring is called a field. That is, in a field, every element has an inverse.
Definition. A subset \(S\) of ring \(R\) is a subring if given any \(r, s \in S\), then \(rs \in S\) and \(r - s \in S\).
Section 16.2 - Integral Domains and Fields
Definition. If \(R\) is a commutative ring and \(r \in R\), then \(r\) is said to be a zero divisor if there is some nonzero \(s \in R\) such that \(rs = 0\).
Definition. A commutative ring with no zero divisors is called an integral domain.
Example. Consider the set \(\mathbb{Z}[i] = \{m + ni | m, n \in \mathbb{Z}\}\). This ring is called the Gaussian integers. Prove that the Gaussian integers are not a field, and are an integral domain.
Example. Proposition 16.15: Cancellation law. Let \(D\) be a commutative ring with identity. Then, \(D\) is an integral domain if and only if for every nonzero \(a \in R\), \(ab = ac\) implies \(b = c\).
Theorem. 16.16: Every finite integral domain is a field.
Definition. For any non-negative integer \(n \in \mathbb{N}\) and \(r \in R\), we say that \(nr = r + \ldots + r \text{(n times)}\).
Definition. The characteristic of a ring is the least possible \(n \in \mathbb{N}\) such that \(nr = 0\) for all \(r \in R\).
Example. For every prime number \(p\), \(\mathbb{N}_p\) is a field of characteristic \(p\).
Lemma. 16.18: Given \(R\) is a ring with identity, the characteristic of \(1\) is the characteristic of the field.
Theorem. 16.19: The characteristic of an integral domain is prime or zero.
Section 16.3 - Ring Homomorphisms and Ideals
Definition. Given rings \(R\) and \(S\), and a mapping \(\varphi: R \rightarrow S\), we say that \(\varphi\) is a ring homomorphism if the following are satisfied for all elements of \(R\):
Definition. If \(\varphi\) is one-to-one and onto, it is an isomorphism.
Definition. For any ring homomorphism \(\varphi\), the kernel of \(\varphi\) is the set
Definition. Proposition 16.22: Let \(\varphi: R \rightarrow S\) be a ring homomorphism. Then,
- If \(R\) is a commutative ring, then \(\varphi(R) \subseteq S\) is a commutative ring.
- \(\varphi(0_R) = 0_S\)
- Let \(1_R\) and \(1_S\) be the identities in \(R\) and \(S\). If \(\varphi\) is onto, then \(\varphi(1_R) = 1_S\)
- If \(R\) is a field an \(\varphi(R) \neq \{0\}\), then \(\varphi(R) \subseteq S\) is a field.
Definition. A subring \(I \subseteq R\) is asn ideal of \(R\) if, when given \(a \in I, r \in R\), then \(ar\) and \(ra\) are both in \(I\). That is, \(rI \subseteq I\) and \(Ir \subseteq I\).
Definition. Given a commutative ring \(R\) with identity, and \(r \in R\), the set
is an ideal in \(R\). Specifically, \(\langle a \rangle\) is a principal ideal.
Example. Theorem 16.25. Every ideal in \(\mathbb{Z}\) is a principal ideal.
Example. With \(\varphi: R \rightarrow S\), \(\ker \varphi\) is an ideal of \(R\).
Remark. 16.28: We are working with two-sided ideals. If rings are not commutative, we may deal with left ideals and right ideals.
Theorem. 16.29: Let \(I\) be an ideal of \(R\). Then, the factor/quotient ring \(R/I\) is a ring with multiplication defined by
Theorem. 16.30: Let \(I\) be an ideal of \(R\). Then, the map \(\varphi: R \rightarrow R/I\) defined by \(\varphi(r) = r + I\) is a ring homomorphism of \(R\) onto \(R/I\) with \(\ker \varphi = I\).
Theorem. 16.31, First Isomorphism Theorem. Let \(\psi: R \rightarrow S\). Then, \(\ker \psi\) is an ideal of \(R\). Consider the isomorphism \(\varphi: R \rightarrow R/\ker \psi\). There exists an isomorphism \(\eta: R / \ker \psi \rightarrow \psi(R)\) such that \(\psi = \eta \varphi\).
Theorem. 16.32, Second Isomorphism Theorem. Let \(I\) be a subring of \(R\) and \(J\) be an ideal of \(R\). Then, \(I \cap J\) is an ideal of \(I\) and
Theorem. 16.33, Third Isomorphism Theorem. Let \(R\) be a ring and \(I, J\) be ideals of J. If \(J \subsetneq I\), then
Theorem. 16.34, Correspondence Theorem. Let \(I\) be an ideal of \(R\). Then, \(S \mapsto S/I\) is a one-to-one correspondence between the set of subrings \(S\) containing \(I\) (that is, \(I \in S\)) and the set of subrings of \(R/I\). Furthermore, the ideals of \(R\) containing \(I\) correspond to the ideals of \(R/I\).
Section 16.4 - Maximal and Prime Ideals
Definition. Consider ring \(R\) and proper ideal \(M \subseteq R\). Then, \(M\) is a maximal ideal of \(R\) if the ideal \(M\) is not a subset of any ideal except \(R\) itself. That is, given any ideal \(I\) properly containing \(M\), \(I = R\).
Theorem. 16.35: Given a commutative ring with identity \(R\), \(M\) is a maximal ideal if and only if \(R/M\) is a field.
Definition. Consider ring \(R\) and proper ideal \(P \subseteq R\). Then, \(P\) is a prime ideal if given \(ab \in P\), either \(a \in P\) or \(b \in P\).
Theorem. 16.38: Let \(R\) be a commutative ring with identity \(1\). Then, \(P \subseteq R\) is a prime ideal of \(R\) if and only if \(R/P\) is a field.
Let us assume that \(P\) is an ideal in \(R\) and \(R/P\) is an integral domain. Take two elements \(ab \in P\). Now, consider \(a + P\) and \(b + P\) in \(R/P\) such that \((a+P)(b+P) = 0+P = P\). As \(R/P\) is a field, either \(a + P = 0 + P = P\) or \(b + P = 0 + P = P\), meaning either \(a \in P\) or \(b \in P\). Thus, \(P\) is as prime ideal.
Now, assume the opposite. Let \(P\) be prime. Now, we want to show that \(R/P\) is an integral domain.
Consider two elements \(a + P\), \(b + P\) in \(R/P\). We know that
Thus, \(ab \in P\). By symnetry, assume \(a \notin P\). Thus, \(b \in P\) by the definition of a prime ideal, so \(b + P = 0 + P\), meaning \(R/P\) is an integral domain.
Theorem. 16.40: In a commutative ring with identity, every maximal ideal is also a prime ideal.
Section 16.5 - Applications to Computer Science
Lemma. Let \(m, n \in \mathbb{B}\) be given. Then, for any \(a, b \in \mathbb{Z}\), there exists some \(x\) that satisfies
Theorem. Chinese Remainder Theorem. Let \(n_1, \ldots, n_k \in \mathbb{N}\) be given such that \(\gcd(n_i, n_j) = 1\). Then, for any integers \(a_1, \ldots, a_k\), the system
has a solution. Additionally, all systems are congruent modulo \(n_1 n_2 \ldots n_k\).