Ring Isomorphism Checker
Ring Isomorphism Checker determines if the rings \\(\mathbb{Z}/n\mathbb{Z}\\) and \\(\mathbb{Z}/m\mathbb{Z}\\) are isomorphic by comparing their orders. It also visualizes their addition and multiplication tables.
Ring Isomorphism Overview
Two rings are isomorphic if there exists a bijective ring homomorphism between them, preserving both addition and multiplication. For the quotient rings \\(\mathbb{Z}/n\mathbb{Z}\\) and \\(\mathbb{Z}/m\mathbb{Z}\\), they are isomorphic if and only if \\(n = m\\), as the ring \\(\mathbb{Z}/n\mathbb{Z}\\) has order \\(n\\) and characteristic \\(n\\).
Definition: A ring isomorphism from \\(\mathbb{Z}/n\mathbb{Z}\\) to \\(\mathbb{Z}/m\mathbb{Z}\\) is a bijective map \\(\phi\\) such that \\(\phi(a + b) = \phi(a) + \phi(b)\\) and \\(\phi(a \cdot b) = \phi(a) \cdot \phi(b)\\).
Condition: \\(\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/m\mathbb{Z}\\) if and only if \\(n = m\\).
Characteristic: The characteristic of \\(\mathbb{Z}/n\mathbb{Z}\\) is \\(n\\), the smallest positive integer such that \\(n \cdot 1 = 0\\).
Example Calculations
Example 1: \\(n = 6\\), \\(m = 6\\)
Orders: Both rings have order 6.
Characteristic: Both have characteristic 6.
Result: \\(\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/6\mathbb{Z}\\), as \\(n = m\\).
Example 2: \\(n = 4\\), \\(m = 6\\)
Orders: \\(\mathbb{Z}/4\mathbb{Z}\\) has order 4, \\(\mathbb{Z}/6\mathbb{Z}\\) has order 6.
Characteristic: 4 and 6, respectively.
Result: \\(\mathbb{Z}/4\mathbb{Z} \not\cong \mathbb{Z}/6\mathbb{Z}\\), as \\(n \neq m\\).
Example 3: \\(n = 5\\), \\(m = 5\\)
Orders: Both have order 5.
Characteristic: Both have characteristic 5.
Result: \\(\mathbb{Z}/5\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}\\), as \\(n = m\\).