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Modular Integer Ring \(\mathbb{Z}_p\)

The ModularSemiring in algebrax.semiring represents the modular integer ring \(\mathbb{Z}_p = (\{0, 1, \dots, p-1\}, +\bmod p, \times\bmod p, 0, 1)\).

Ring Operations

  • Additive Identity: 0
  • Multiplicative Identity: 1 % p
  • Addition: \((a + b) \bmod p\)
  • Multiplication: \((a \times b) \bmod p\)
  • Exponentiation: \(a^n \bmod p\)

It serves as a foundational coefficient ring for finite field representations (\(\text{GF}(p^m)\)) and modular matrix arithmetic.

Python Example

import algebrax as ax

# Create Z_5 modular ring
z5 = ax.semiring.ModularSemiring(p=5)

# Ring operations in Z_5
print("3 + 4 mod 5:", z5.add(3, 4))    # 2
print("3 * 4 mod 5:", z5.mul(3, 4))    # 2
print("2^4 mod 5:  ", z5.power(2, 4))  # 1

# Matrix multiplication over Z_5
A = {0: {0: 3, 1: 4}, 1: {0: 2, 1: 1}}
B = {0: {0: 2, 1: 1}, 1: {0: 4, 1: 3}}

C = ax.matrix.dot(A, B, semiring=z5)
print("A @ B over Z_5:", C)