Rings, Fields, and Integral Domains

While group theory analyzes systems with a single operation, ring theory deals with sets that have two interconnected binary operations, typically identified as addition ( $+$ ) and multiplication ( $\cdot$ ).

Defining a Ring

A ring is a set $R$ equipped with these two operations that must adhere to three specific layers of axioms:

Addition forms an Abelian group: Under the operation of addition, the set must be associative and commutative. It must contain an additive identity ( $0$ ), and every element $a$ must have an additive inverse ( $-a$ ).
Multiplication forms a Monoid: Under multiplication, the set must be associative. Furthermore, it must possess a multiplicative identity ( $1$ ) such that multiplying any element by $1$ leaves it unchanged.
Distributivity: Multiplication must distribute over addition from both the left and the right. This means $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ , and similarly $(b + c) \cdot a = (b \cdot a) + (c \cdot a)$ .

Interactive Lab

def is_ring_add_abelian(elements, add_op, zero):
  for a in elements:
      if add_op(a, zero) != a: return False
      for b in elements:
          if add_op(a, b) != add_op(b, a): return False
          if add_op(a, b) not in elements: return False
  return True

# Z/4Z Addition
elements = [0, 1, 2, 3]
add = lambda a, b: (a + b) % 4
print(f"Is addition in Z/4Z Abelian? {is_ring_add_abelian(elements, add, 0)}")

python

Interactive Lab

Read the code, make a small change, then run it and inspect the output. Runtime setup messages stay outside the terminal so the result remains focused on what the program prints.

Step 1

Inspect the idea

Step 2

Edit the program

Step 3

Run and compare

Integral Domains

A domain is defined as a nonzero ring that lacks nonzero zero-divisors—meaning you cannot multiply two nonzero elements together and get zero as a result.

Building upon this, an Integral Domain is simply a domain that is also commutative. Thus, it is a commutative, nonzero ring where the product of any two nonzero elements is guaranteed to be nonzero. This property is what allows basic cancellation laws to function in algebraic equations.

Interactive Lab

def find_zero_divisors(elements, mult_op, zero):
  divisors = []
  for a in elements:
      if a == zero: continue
      for b in elements:
          if b == zero: continue
          if mult_op(a, b) == zero:
              divisors.append(a)
              break
  return divisors

# Z/6Z multiplication
mult = lambda a, b: (a * b) % 6
z6 = [0,1,2,3,4,5]
print(f"Zero divisors in Z/6Z: {find_zero_divisors(z6, mult, 0)}")

python

Interactive Lab

Read the code, make a small change, then run it and inspect the output. Runtime setup messages stay outside the terminal so the result remains focused on what the program prints.

Step 1

Inspect the idea

Step 2

Edit the program

Step 3

Run and compare

Principal Ideal Domains (PID)

In ring theory, an ideal is a subset of a ring that absorbs multiplication by any ring element. An ideal is considered “principal” if it can be entirely generated by multiples of a single element from the ring.

A Principal Ideal Domain (PID) is defined strictly as an integral domain wherein every single ideal is a principal ideal. The set of integers ( $\mathbb{Z}$ ) is the classic example of a PID.

In Z/12Z, is the element 5 a unit?

References & Further Reading

Wikipedia: Ring (mathematics) (CC-BY-SA 3.0)