Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| group |
✓ |
The integers modulo $5$ form a group under addition modulo $5$, because $0$ is an identity and $5 - n$ is the inverse of $n$.
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| ring |
✓ |
The integers modulo $5$, with addition and multiplication modulo $5$, form a ring, because the rules of integer arithmetic still hold after taking remainders.
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| field |
✓ |
The integers modulo $5$ form a field, because every nonzero element has a multiplicative inverse: $1 \cdot 1 = 1$, $2 \cdot 3 = 6 = 1$, and $4 \cdot 4 = 16 = 1$ modulo $5$.
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Distinctions using this example
- Group vs. ring vs. field: The integers modulo $5$ form a field, but the integers modulo $6$ do not, because $5$ is prime and $6$ is not.
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