Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| group |
✓ |
The integers modulo $6$ form a group under addition modulo $6$, because $0$ is an identity and $6 - n$ is the inverse of $n$.
|
| ring |
✓ |
The integers modulo $6$, with addition and multiplication modulo $6$, form a ring, because the rules of integer arithmetic still hold after taking remainders.
|
| field |
✗ |
The integers modulo $6$ do not form a field, because $2$ has no multiplicative inverse: $2 k$ modulo $6$ is $0$, $2$, or $4$, and none of these is $1$. Also, $2 \cdot 3 = 0$ modulo $6$, although neither $2$ nor $3$ is $0$.
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