Abstract Algebra - Ring

Ring

By a ring we mean a set \(A\) with operations called addition and multiplication which satisfy the following axioms:

• \(A\) with addition alone is an abelian group.

• Multiplication is associative.

• Multiplication is distributive over addition. That is, for all \(a, b\) and \(c\) in \(A\), we have

  \[ a(b+c) = ab + ac \\ (b+c)a = ba + ca \]

💡 When multiplication is communicative in a ring, we call that ring a communicative ring.

💡 When there is a neutral element for multiplication, then that neutral element is called the unity of the ring.

Basic Examples of Ring

• \(\mathbb{Z}\) is the ring of the integers.
• \(\mathbb{Q}\) is the ring of the rational numbers.
• \(\mathbb{R}\) is the ring of the real numbers.
• \(\mathbb{C}\) is the ring of the complex numbers.
• \(\mathscr{F}(\mathbb{R})\) is the ring of real functions.

Trivial Ring

A ring with only the zero element \(\{ 0 \}\) is called the trivial ring.

Divisor of Zero

In any ring, a nonzero element \(a\) is called a divisor of zero (除数，零因子) if there is a nonzero element \(b\) in the ring such that the product \(ab\) or \(ba\) is equal to zero. 

💡 For example, 2 and 3 are divisors of zero in \(\mathbb{Z}_6\).

Cancellation Property

A ring is said to have the cancellation property if

\[ ab = ac \text{ or } ba = ca \implies b = c \]

for any elements \(a, b, c\) in the ring and \(a \neq 0\).

💡 A ring has the cancellation property iff it has no divisors of zero.

Direct Product

If \(A\) and \(B\) are rings, their direct product is a new ring, denoted by \(A \times B\), and defined as follows: \(A \times B\) consists of all the ordered pairs \((x, y)\) where \(x \in A\) and \(y \in B\). Addition in \(A \times B\) consists of adding corresponding components:

\[ (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2) \]

Multiplication in \(A \times B\) consists of multiplying corresponding components:

\[ (x_1, y_1) \cdot (x_2, y_2) = (x_1y_1, x_2y_2) \]

Subring

If a nonempty subset \(B \sube A\) is closed with respect to addition, multiplication, and negatives, then \(B\) with the operations of \(A\) is a subring of \(A\).

Ideal

A nonempty subset \(B\) of a ring \(A\) is called an ideal of \(A\) if \(B\) is closed with respect to addition and negative, and \(B\) absorbs products in \(A\).

💡 \(B\) absorbs products in \(A\) if for all \(b \in B\) and \(x \in A\), \(xb\) and \(bx\) are in \(B\).

Ideals are in rings as normal subgroups are in groups.

Principal ideal

In a commutative ring with unity, the set of all the multiples of a fixed element \(a\) by all the elements in the ring

\[ xa \]

is called the principal ideal generated by \(a\), and is denoted by \(\langle a \rangle\).

Coset of Rings

Let \(A\) be a ring, and \(J\) an ideal of \(A\). For any element \(a \in A\), the symbol \(J + a\) denotes the set of all sums \(j + a\), as \(a\) remains fixed and \(j\) ranges over \(J\). That is,

\[ J + a = \{ j + a: j \in J\} \]

\(J + a\) is called a coset of \(J\) in \(A\).

Coset Addition and Multiplication

Additions and multiplications of coset of rings are as follows:

\[ (J + a)+(J + b) = J + (a+b) \\ (J + a)(J + b) = J + ab \]

Quotient Rings

\(A/J\) with coset addition and multiplication is a ring, and it is called quotient ring.

💡 \(A/J\) is a homomorphic image of \(A\).

Integral domains

An integral domain is defined to be a communicative ring with unity have the cancellation property.

Characteristic

The characteristic of \(A\) is the least positive integer \(n\) such that

\[ \underbrace{1 + 1 + \cdots + 1}_{n \text{ times }} = 0 \]

If there is no such positive integer \(n\), \(A\) has characteristic \(0\).

Theorem 1

All the nonzero elements in an integral domain have the same additive order.

💡 which is equal to the characteristic of \(1\). Since \(n \cdot a = a + a + \cdots + a = (n \cdot 1)a\). Since \(a \neq 0\), then \(n \cdot 1 = 0\) makes \(n \cdot a = 0\).

Theorem 2

In an integral domain with nonzero characteristic, the characteristic is a prime number.

Theorem 3

In any integral domain of characteristic \(p\), for all elements \(a\) and \(b\),

\[ (a+b)^p = a^p + b^p \]

Theorem 4

Every finite integral domain is a field.

Ordered integral domain

An ordered inttegral domain is an integral domain \(A\) with a relation, symbolized by \(<\), having the following properties:

1. For any \(a\) and \(b\) in \(A\), exactly one of the following is true: \(a = b, a < b, a > b\).
2. If \(a < b\) and \(b < c\), then \(a < c\).
3. If \(a < b\), then \(a + c < b + c\).
4. If \(a < b\), then \(ac < bc\) on the condition that \(0 < c\).

Integral System

An ordered integral domain \(A\) is called an integral system if every nonempty subset of \(A^+\) has a least element.

Theorem 1

Let \(K\) represent a set of positive integers. Consider the following two conditions:

1. \(1 \in K\).
2. For any positive integer \(k\), if \(k \in K\), then \(k + 1 \in K\).

If \(K\) is any set of positive integers satisfying these two conditions, then \(K\) consists of all the positive integers.

Theorem 2: Principle of mathematical induction

1. \(S_1\) is true.
2. For any positive integer \(k\), if \(S_k\) is true, then \(S_{k+1}\) is true.

If conditions 1 and 2 are satisfied, then \(S_n\) is true for every positive iinteger \(n\).

Theorem 3: Division algorithm

If \(m\) and \(n\) are integers and \(n\) is positive, there exist unique integers \(q\) and \(r\) such that

\[ m = nq + r \text{ and } 0 \leq r < n \]

💡 We call \(q\) the quotient, and \(r\) the remainder, in the division of \(m\) by \(n\).

Factoring into Primes

Theorem 1

Every ideal of \(\mathbb{Z}\) is principal.

Theorem 2

The only invertible elements of \(\mathbb{Z}\) are 1 and -1.

💡 A pair of integers \(r\) and \(s\) are called associates if they divide each other. If \(r\) and \(s\) are associates in \(\mathbb{Z}\), then \(r = \pm s\).

Theorem 3

Any two nonzero integers \(r\) and \(s\) have a greatest common divisor \(t\). Furthermore, \(t\) is equal to a "linear combination" of \(r\) and \(s\). That is

\[ t = kr + ls \]

for some integers \(k\) and \(l\).

💡 A greatest common divisor of \(r\) and \(s\) is an integer \(r\) such that

1. \(t | r\) and \(t|s\).
2. For any integer \(u\), if \(u | r\) and \(u | s\), then \(u | t\).

which is denoted by \(t = \gcd(r, s)\)

Composite number lemma

If a positive integer \(m\) is composite, then \(m = rs\) where

\[ 1< r< m \text{ and } 1<s<m \]

Euclid's lemma

Let \(m\) and \(n\) be integers, and let \(p\) be a prime. If \(p | (mn)\), then either \(p|m\) or \(p|n\).

Corollary 1

Let \(m_1, \dots , m_t\) be integers, and let \(p\) be a prime. If \(p|(m_1 \cdots m_t)\), then \(p|m_i\) for on of the factors \(m_i\) among \(m_1, \dots, m_t\).

Corollary 2

Let \(q_1, \dots, q_t\) and \(p\) be positive primes. If \(p | (q_1 \dots q_t)\), then \(p\) is equal to one of the factors \(q_1, \dots, q_t\).

Factorization into primes

Every integer \(n > 1\) can be expressed as a product of positive primes. That is, there are one or more primes \(p_1, \dots, p_r\) such that

\[ n = p_1p_2 \cdots p_r \]

And the factorization is unique.