= Homomorphism
{title2=$f:A\to B$}
{wiki}
A homomorphism is a map between <algebraic structures> preserving their specified operations. For a single <binary operation>, this means $f(a\ast b)=f(a)\star f(b)$, where the source and target may use different operation symbols. A <group homomorphism> preserves the group operation; a <ring homomorphism> preserves addition and multiplication, with the identity convention specified; a <module homomorphism> preserves addition and scalar multiplication.
Reduction modulo $5$ is a <ring homomorphism> $\mathbb Z\to\mathbb Z/5\mathbb Z$: adding or multiplying first and then reducing gives the same residue as reducing first and then operating. For example, $12\cdot14=168\equiv3\pmod5$, while the reduced operands give $2\cdot4=8\equiv3\pmod5$. This lets a complicated integer calculation be projected into simpler finite arithmetic. It also explains why a <polynomial equation> over the <integers> imposes necessary <modular arithmetic> conditions on any integer solution.
A homomorphism can lose information: $0$ and $5$ have the same image under that reduction. An <isomorphism> loses none of the structure because it has an operation-preserving inverse. Preservation of the operation, rather than preservation of arbitrary labels or sizes, is the essential condition.
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