math / Structures / free
Ring Theory
Compare addition and multiplication in finite systems where zero can change the rules.
Follow the mechanism1 / 3
One set carries addition and multiplication. Residues can be combined in two ways, and distributivity links those operations without making their inverse behavior identical.
finite ringZ/8Z · × · selected 2
Focus the idea
Change the modulus
| × | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 0 | 2 | 4 | 6 | 0 | 2 | 4 | 6 |
| 3 | 0 | 3 | 6 | 1 | 4 | 7 | 2 | 5 |
| 4 | 0 | 4 | 0 | 4 | 0 | 4 | 0 | 4 |
| 5 | 0 | 5 | 2 | 7 | 4 | 1 | 6 | 3 |
| 6 | 0 | 6 | 4 | 2 | 0 | 6 | 4 | 2 |
| 7 | 0 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
Inspect element
additive inverse6
multiplicative inversenone
zero divisors3
field?no
Selected element2
zero divisor in Z/8Z.
Addition2 + 6 = 0
Every element has a partner that returns to the additive identity.
Multiplication2 × 2 = 4
It meets 4 at zero.
2 × 4 ≡ 0 (mod 8), so 2 is a zero divisor rather than a unit.
A ring carries two operations at once: addition behaves like a reversible group, while multiplication distributes over it but may contain non-units and zero divisors. That tension is what makes ring structure richer than a single operation table.