= Reduction map on unit groups
{title2=$R^\times\to(R/I)^\times$}
{wiki}
Reduction modulo an <ideal> $I$ of a <commutative ring> $R$ gives a <group homomorphism> $R^\times\to(R/I)^\times$. It need not be surjective: the <units> $\{1,-1\}$ of $\mathbb Z$ do not lift the classes two and three in $(\mathbb Z/5\mathbb Z)^\times$.
A useful sufficient condition is $I\subseteq J(R)$, the <Jacobson radical>. If $ab\equiv1\pmod I$, then $ab$ is a <unit> by the <unit criterion for the Jacobson radical>, and consequently $a$ is a <unit>. Every invertible residue class then lifts.
For positive <integers> $n\mid m$, reduction $(\mathbb Z/m\mathbb Z)^\times\to(\mathbb Z/n\mathbb Z)^\times$ is always surjective. Start with a representative coprime to $n$ and use the <Chinese remainder theorem> to choose a congruent representative avoiding each additional prime dividing $m$. This explicit <integer> result should be distinguished from unrestricted reduction in an arbitrary <ring>.
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