Rings

In abstract algebra, a ring is a fundamental algebraic structure. It is a set equipped with two binary operations, usually called addition and multiplication, that generalize the familiar arithmetic operations of integers. For a set R to be a ring, it must satisfy certain axioms. These axioms ensure that the operations behave in a predictable and structured way.

Axioms of a Ring

Let R be a non-empty set and let '+' and '·' be two binary operations on R. R is called a ring if the following axioms hold:

1. (R, +) is an abelian group. This means:
   • Closure under addition: For all a, b in R, a + b is in R.
   • Associativity of addition: For all a, b, c in R, (a + b) + c = a + (b + c).
   • Existence of additive identity: There exists an element 0 in R such that for all a in R, a + 0 = 0 + a = a.
   • Existence of additive inverse: For every a in R, there exists an element -a in R such that a + (-a) = (-a) + a = 0.
   • Commutativity of addition: For all a, b in R, a + b = b + a.
2. (R, ·) is closed under multiplication: For all a, b in R, a · b is in R.
3. Multiplication is associative: For all a, b, c in R, (a · b) · c = a · (b · c).
4. Multiplication distributes over addition: For all a, b, c in R:
   • Left distributive law: a · (b + c) = (a · b) + (a · c).
   • Right distributive law: (b + c) · a = (b · a) + (c · a).

Types of Rings

Rings can be further classified based on additional properties they might possess. Some important types include:

• Commutative Ring: A ring R is commutative if multiplication is commutative, i.e., for all a, b in R, a · b = b · a. The integers (Z) form a commutative ring.
• Ring with Unity (or Ring with Identity): A ring R is said to have unity if there exists an element 1 in R such that for all a in R, a · 1 = 1 · a = a. The integers (Z) with unity are a common example.
• Integral Domain: A commutative ring with unity (1 ≠ 0) is an integral domain if it has no zero divisors. This means that if a · b = 0, then either a = 0 or b = 0. The integers (Z) are an integral domain.
• Field: A commutative ring with unity (1 ≠ 0) is a field if every non-zero element has a multiplicative inverse. That is, for every a ∈ R, a ≠ 0, there exists an element a^-1 ∈ R such that a · a^-1 = a^-1 · a = 1. Fields are integral domains, and every field is a commutative ring with unity. The set of rational numbers (Q) and real numbers (R) are fields.

Example of a Ring

Consider the set of 2x2 matrices with real entries: $$ M_2(\mathbb{R}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid a, b, c, d \in \mathbb{R} \right\} $$ with the usual matrix addition and multiplication.

• Addition of matrices is component-wise, satisfying all abelian group axioms.
• Matrix multiplication is defined as: $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix} $$ This operation is closed and associative.
• Matrix multiplication distributes over addition.

However, $M_2(\mathbb{R})$ is not a commutative ring because matrix multiplication is generally not commutative. For instance: $$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \quad \text{but} \quad \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} $$ It is a ring with unity, where the unity is the identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. It is also not an integral domain because it has zero divisors (e.g., $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$).

Euclidean Rings

A Euclidean ring is a special type of integral domain that has a structure similar to the integers, allowing for a division algorithm. This property is crucial for many number-theoretic algorithms, such as the Euclidean algorithm for finding the greatest common divisor (GCD).

Definition of a Euclidean Ring

An integral domain R is called a Euclidean ring if there exists a function $d: R \setminus \{0\} \to \mathbb{N}_0$ (where $\mathbb{N}_0 = \{0, 1, 2, ...\}$) such that the following two conditions are met:

1. Division Algorithm Property: For any two elements a, b in R with b ≠ 0, there exist elements q (quotient) and r (remainder) in R such that $a = bq + r$, where either $r = 0$ or $d(r) < d(b)$.
2. Minimal Property: For any non-zero element a in R, $d(a) \geq 1$. (Some definitions include $d(0)=0$ and define $d$ on all of R).

The function $d$ is often referred to as the "degree function" or "Euclidean function."

Properties and Examples of Euclidean Rings

• Integers (Z): The set of integers Z is a Euclidean ring with the degree function $d(n) = |n|$ (the absolute value of n). For any integers a and b ≠ 0, we can find integers q and r such that $a = bq + r$ with $0 \leq r < |b|$. This is the standard division algorithm for integers.
• Polynomials over a Field (F[x]): The ring of polynomials in one variable x with coefficients from a field F, denoted by F[x], is a Euclidean ring. The degree function is the usual degree of the polynomial, $d(p(x)) = \deg(p(x))$. For any polynomials a(x) and b(x) in F[x] with b(x) ≠ 0, there exist unique polynomials q(x) and r(x) such that $a(x) = b(x)q(x) + r(x)$, where $r(x) = 0$ or $\deg(r(x)) < \deg(b(x))$.
• Gaussian Integers (Z[i]): The set of Gaussian integers, $Z[i] = \{a + bi \mid a, b \in Z\}$, forms a Euclidean ring. The degree function is $d(a + bi) = a^2 + b^2$, which is the square of the modulus of the complex number $a+bi$.
• Eisenstein Integers (Z[ω]): The set of Eisenstein integers, $Z[\omega] = \{a + b\omega \mid a, b \in Z\}$, where $\omega = e^{2\pi i/3} = \frac{-1 + i\sqrt{3}}{2}$, also forms a Euclidean ring. The degree function is $d(a + b\omega) = a^2 - ab + b^2$, which is the square of the norm.

Significance of Euclidean Rings

• Existence of GCD: In a Euclidean ring, the greatest common divisor (GCD) of any two non-zero elements exists and can be found using the Euclidean algorithm.
• Principal Ideal Domains (PIDs): Every Euclidean ring is a Principal Ideal Domain (PID). This means that every ideal in a Euclidean ring is generated by a single element.
• Unique Factorization Domains (UFDs): Every PID is a Unique Factorization Domain (UFD). Therefore, every Euclidean ring is also a UFD. This property is fundamental in number theory and algebra.

The Euclidean Algorithm

The Euclidean algorithm is a method for efficiently computing the GCD of two integers. It relies on the division algorithm. To find GCD(a, b) where a > b > 0:

1. Divide a by b to get $a = bq_1 + r_1$, where $0 \leq r_1 < b$.
2. If $r_1 = 0$, then GCD(a, b) = b.
3. If $r_1 \neq 0$, divide b by $r_1$ to get $b = r_1q_2 + r_2$, where $0 \leq r_2 < r_1$.
4. If $r_2 = 0$, then GCD(a, b) = $r_1$.
5. If $r_2 \neq 0$, continue this process: $r_{i-1} = r_i q_{i+1} + r_{i+1}$, until a remainder $r_{k+1} = 0$ is obtained.
6. The last non-zero remainder, $r_k$, is the GCD(a, b).

This algorithm is directly applicable in any Euclidean ring by using the division property with the degree function.

Polynomial Rings

A polynomial ring is a ring formed by polynomials with coefficients from some base ring or field. These rings are fundamental in algebra and have applications in many areas, including coding theory, cryptography, and algebraic geometry.

Definition of a Polynomial Ring

Let R be a commutative ring with unity. The polynomial ring in one indeterminate x over R, denoted by R[x], is the set of all polynomials of the form: $$ p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 $$ where $a_i \in R$ for all $i$, and $n$ is a non-negative integer. The elements $a_i$ are called the coefficients of the polynomial.

The operations of addition and multiplication in R[x] are defined as follows:

• Addition: To add two polynomials, we add their corresponding coefficients. If $p(x) = \sum_{i=0}^n a_i x^i$ and $q(x) = \sum_{i=0}^m b_i x^i$, then $p(x) + q(x) = \sum_{i=0}^{\max(n,m)} (a_i + b_i) x^i$, where we consider coefficients of missing terms to be zero.
• Multiplication: To multiply two polynomials, we use the distributive law and the rule $x^i \cdot x^j = x^{i+j}$. If $p(x) = \sum_{i=0}^n a_i x^i$ and $q(x) = \sum_{j=0}^m b_j x^j$, then $p(x) \cdot q(x) = \sum_{k=0}^{n+m} c_k x^k$, where $c_k = \sum_{i+j=k} a_i b_j$.

Properties of Polynomial Rings

• R[x] is a commutative ring if R is a commutative ring.
• R[x] has unity if R has unity. The unity is the constant polynomial 1.
• If R is an integral domain, then R[x] is also an integral domain.
• Degree of a Polynomial: If $p(x) = a_n x^n + \dots + a_0$ with $a_n \neq 0$, then the degree of p(x), denoted by deg(p(x)), is n. The degree of the zero polynomial is usually defined as $-\infty$ or undefined.
• Degree of Product: If R is an integral domain, then for non-zero polynomials p(x) and q(x), $\deg(p(x)q(x)) = \deg(p(x)) + \deg(q(x))$.

Polynomial Rings over Fields

When the base ring R is a field F, the polynomial ring F[x] has particularly important properties.

• Euclidean Domain: As mentioned earlier, F[x] is a Euclidean ring with the degree function $d(p(x)) = \deg(p(x))$. This is a crucial result that allows for polynomial division with remainder, similar to integer division.
• Principal Ideal Domain (PID): Since F[x] is a Euclidean ring, it is also a PID. Every ideal in F[x] is generated by a single polynomial. For example, the ideal generated by $x^2 - 1$ is $(x^2 - 1) = \{ (x^2 - 1)q(x) \mid q(x) \in F[x] \}$.
• Unique Factorization Domain (UFD): Consequently, F[x] is also a UFD. Polynomials in F[x] can be factored into irreducible polynomials in essentially a unique way, analogous to prime factorization of integers.

Example: Polynomials over Real Numbers (R[x])

Consider the polynomial ring R[x].

• Let $p(x) = 3x^3 - 2x + 5$ and $q(x) = x^2 + 1$. Both polynomials are in R[x].
• Addition: $p(x) + q(x) = (3x^3 - 2x + 5) + (x^2 + 1) = 3x^3 + x^2 - 2x + 6$.
• Multiplication: $$ p(x)q(x) = (3x^3 - 2x + 5)(x^2 + 1) $$ $$ = (3x^3)(x^2) + (3x^3)(1) + (-2x)(x^2) + (-2x)(1) + (5)(x^2) + (5)(1) $$ $$ = 3x^5 + 3x^3 - 2x^3 - 2x + 5x^2 + 5 $$ $$ = 3x^5 + x^3 + 5x^2 - 2x + 5 $$
• Degree: $\deg(p(x)) = 3$, $\deg(q(x)) = 2$. $\deg(p(x)q(x)) = 5$, which is $3 + 2$.
• Division Algorithm: We can divide $p(x)$ by $q(x)$. $$ 3x^3 - 2x + 5 = (x^2 + 1)(3x) + (-5x + 5) $$ Here, the quotient is $q_1(x) = 3x$ and the remainder is $r_1(x) = -5x + 5$. The degree of the remainder is 1, which is less than the degree of the divisor (2).

Multiple Indeterminates

We can also form polynomial rings with multiple indeterminates. For example, R[x, y] is the ring of polynomials in two variables x and y with coefficients in R. The elements are sums of terms like $c x^i y^j$, where $c \in R$.

For example, $3x^2y - 5xy^3 + 7$ is an element of R[x, y].

R[x, y] is commutative if R is commutative. However, R[x, y] is not a Euclidean ring or a PID in general.

Unique Factorization Domains (UFDs)

A Unique Factorization Domain (UFD) is a fundamental concept in ring theory, particularly in integral domains. It generalizes the idea of unique prime factorization of integers to a broader algebraic setting. In a UFD, every non-zero, non-unit element can be uniquely expressed as a product of irreducible elements, up to the order of the factors and multiplication by units.

Definitions and Key Elements

Let R be an integral domain.

• Units: An element $u \in R$ is a unit if there exists an element $v \in R$ such that $uv = 1$. In the ring of integers Z, the units are 1 and -1. In a field F, every non-zero element is a unit.
• Irreducible Element: A non-zero, non-unit element $p \in R$ is irreducible if whenever $p = ab$ for some $a, b \in R$, then either a or b must be a unit. In the integers Z, the irreducible elements are the prime numbers (and their negatives).
• Prime Element: A non-zero, non-unit element $q \in R$ is prime if whenever $q$ divides $ab$ for some $a, b \in R$, then $q$ divides a or $q$ divides b.

In any integral domain, every prime element is irreducible. Also, in a commutative ring with unity, every irreducible element is prime if and only if the ring is a UFD.

Definition of a UFD

An integral domain R is called a Unique Factorization Domain (UFD) if it satisfies the following two conditions:

1. Existence of Factorization: Every non-zero, non-unit element in R can be written as a product of a finite number of irreducible elements.
2. Uniqueness of Factorization: If $a = p_1 p_2 \dots p_n$ and $a = q_1 q_2 \dots q_m$ are two factorizations of an element $a$ into irreducible elements, then $n = m$, and after possibly reordering the $q_j$'s, we have $p_i = u_i q_i$ for each $i=1, \dots, n$, where $u_i$ are units in R.

Relationship with Other Ring Properties

UFDs occupy a significant place in the hierarchy of integral domains:

• Every Euclidean Ring is a Principal Ideal Domain (PID).
• Every PID is a Unique Factorization Domain (UFD).

The converse statements are not always true:

• Not every UFD is a PID (e.g., $Z[x]$ is a UFD but not a PID).
• Not every PID is a Euclidean Ring (e.g., $Z[\frac{1+\sqrt{-19}}{2}]$ is a PID but not Euclidean).

Examples of UFDs

• Integers (Z): The set of integers is the classic example of a UFD. Any integer can be uniquely factored into primes (e.g., $12 = 2^2 \cdot 3$).
• Fields (F): Any field is trivially a UFD, as every non-zero element is a unit, and the factorization condition holds vacuously.
• Polynomial Rings over Fields (F[x]): As we've seen, F[x] is a Euclidean ring, hence a PID and a UFD. Polynomials can be factored into unique irreducible polynomials (analogous to primes).
• Polynomial Rings over UFDs (R[x] where R is a UFD): If R is a UFD, then the polynomial ring R[x] is also a UFD. This is a powerful result. For example, since Z is a UFD, Z[x] is a UFD.
• Gaussian Integers (Z[i]): This ring is a Euclidean ring, so it is a UFD. The factorization of Gaussian integers into Gaussian primes is unique.

Example: Factorization in Z[x]

Consider the polynomial ring Z[x]. This ring is a UFD. Let's look at the factorization of $p(x) = x^2 - 4$.

• In Z[x], we can factor this as $p(x) = (x - 2)(x + 2)$. Both $(x-2)$ and $(x+2)$ are irreducible in Z[x].
• The units in Z[x] are just 1 and -1.
• If we try to factor further, say $(x-2) = ab$, then either $a$ or $b$ must be a unit. This confirms that $(x-2)$ is irreducible.

Now consider $q(x) = 2x + 2$.

• We can factor out the integer 2: $q(x) = 2(x+1)$.
• Here, 2 is an irreducible element in Z[x] (it's a prime integer).
• $(x+1)$ is also irreducible in Z[x].
• The factorization $2(x+1)$ is unique up to units and order.

Quotient Rings and Ideals

Quotient rings (also known as factor rings) are constructed by "dividing" a ring by one of its ideals. This process is analogous to how we form quotient groups by dividing a group by one of its normal subgroups, or how we form integers modulo n (Z/nZ) by dividing the integers Z by the ideal nZ. Ideals are the key structures that allow us to form quotient rings.

Ideals

An ideal of a ring R is a non-empty subset I of R that satisfies the following conditions:

1. Closure under Subtraction: For any two elements a, b in I, their difference a - b is also in I. (This implies that (I, +) is a subgroup of (R, +)).
2. Absorption Property: For any element a in I and any element r in R, both $r \cdot a$ and $a \cdot r$ are in I.

Essentially, an ideal is a special kind of subset of a ring that is closed under addition and subtraction and absorbs multiplication by any element of the ring.

Types of Ideals

• Principal Ideal: In a commutative ring R, the principal ideal generated by an element 'a', denoted by (a) or , is the set of all multiples of 'a' by elements of R. That is, $(a) = \{ra \mid r \in R\}$.
• Maximal Ideal: An ideal M in a commutative ring R is called a maximal ideal if M is a proper ideal (M ≠ R) and the only ideal properly containing M is R itself. In other words, if I is an ideal such that $M \subseteq I \subseteq R$, then either $I = M$ or $I = R$.
• Prime Ideal: An ideal P in a commutative ring R is called a prime ideal if P is a proper ideal (P ≠ R) and for any elements a, b in R, if the product $ab \in P$, then either $a \in P$ or $b \in P$.