Rings, Fields, and Integral Domains

While group theory deals with a single binary operation, many of the most important structures in mathematics—such as the set of integers, polynomials, and real numbers—involve two operations: addition and multiplication. Abstract algebra formalizes these through the concepts of Rings and Fields.

The Definition of a Ring

A ring $(R,+,\cdot )$ is a set $R$ equipped with two binary operations, addition and multiplication, satisfying:

1. $(R,+)$ is an Abelian Group: It satisfies closure, associativity, identity ($0$), and inverses ($-a$).
2. Multiplication is Associative: For all $a,b,c\in R$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3. Distributive Laws: Multiplication distributes over addition:
   • $a\cdot (b+c)=a\cdot b+a\cdot c$
   • $(b+c)\cdot a=b\cdot a+c\cdot a$

Commutativity and Units

• A ring is commutative if $a\cdot b=b\cdot a$ for all $a,b\in R$.
• A ring with identity (or unital ring) has a multiplicative identity $1$ such that $1\cdot a=a\cdot 1=a$.
• An element $u\in R$ is a unit if it has a multiplicative inverse ($uv=vu=1$). The set of units forms a group $R^{\times }$.

Integral Domains and Zero Divisors

One key property of the integers $\mathbb{Z}$ is that if $ab=0$, then $a=0$ or $b=0$. This is not true in all rings.

• Zero Divisors: A non-zero element $a\in R$ is a zero divisor if there exists a non-zero $b\in R$ such that $ab=0$.
  • Example: In $\mathbb{Z}/6\mathbb{Z}$, $[2][3]=[0]$, so $[2]$ and $[3]$ are zero divisors.
• Integral Domain: A commutative ring with identity that has no zero divisors. Examples: $\mathbb{Z}$, $\mathbb{R}[x]$.

Ideals and Quotient Rings

Just as normal subgroups are used to form quotient groups, ideals are used to form quotient rings.

Definition of an Ideal

A subset $I\subseteq R$ is a (two-sided) ideal if:

1. $(I,+)$ is a subgroup of $(R,+)$.
2. For all $r\in R$ and $i\in I$, both $ri\in I$ and $ir\in I$. (The ideal “absorbs” multiplication).

Quotient Ring $R/I$

If $I$ is an ideal of $R$, the set of cosets $R/I=\{r+I\mid r\in R\}$ forms a ring under:

• $(a+I)+(b+I)=(a+b)+I$
• $(a+I)(b+I)=(ab)+I$

Prime and Maximal Ideals

• Maximal Ideal: $M\subset R$ is maximal if no ideal $J$ exists such that $M\subset J\subset R$. $R/M$ is a field.
• Prime Ideal: $P\subset R$ is prime if $ab\in P⟹a\in P$ or $b\in P$. $R/P$ is an integral domain.

Euclidean Domains and PIDs

We can refine types of rings based on their factorization properties:

1. Unique Factorization Domain (UFD): Every non-zero, non-unit element has a unique prime factorization (e.g., $\mathbb{Z},F[x]$).
2. Principal Ideal Domain (PID): Every ideal is generated by a single element (e.g., $\mathbb{Z},F[x]$).
3. Euclidean Domain (ED): A PID where a division algorithm exists (e.g., integers with absolute value, polynomials with degree).

Fields

A field is a commutative ring with identity $1\ne 0$ in which every non-zero element is a unit. Fields are the most symmetrical of algebraic structures, enabling addition, subtraction, multiplication, and division (except by zero).

The Characteristic of a Field

The characteristic of a field $F$, denoted $char(F)$, is the smallest positive integer $n$ such that $n\cdot 1=0$, or $0$ if no such $n$ exists.

• $\mathbb{Q},\mathbb{R},\mathbb{C}$ have $char=0$.
• ${\mathbb{F}}_p$ has $char=p$.

Prime Fields

Every field contains a smallest subfield called its prime field.

• If $char(F)=0$, the prime field is isomorphic to $\mathbb{Q}$.
• If $char(F)=p$, the prime field is isomorphic to ${\mathbb{F}}_p$.

Extensions and Modular Polynomials

Just as we construct $\mathbb{Z}/n\mathbb{Z}$, we can construct fields by quotienting polynomial rings by irreducible polynomials. This is how we define the field of complex numbers: $\mathbb{R}[x]/⟨x^2+1⟩\cong \mathbb{C}$

Python: Modeling Quotient Rings

class ModuloRing:
    def __init__(self, n):
        self.n = n

    def add(self, a, b):
        return (a + b) % self.n

    def multiply(self, a, b):
        return (a * b) % self.n

    def is_integral_domain(self):
        # Check for zero divisors
        for a in range(1, self.n):
            for b in range(1, self.n):
                if self.multiply(a, b) == 0:
                    return False, (a, b)
        return True, None

# Z/5Z is an integral domain (and a field)
# Z/6Z is not
for n in [5, 6]:
    ring = ModuloRing(n)
    is_id, divisors = ring.is_integral_domain()
    print(f"Z/{n}Z is integral domain: {is_id}")

Exercise