A $m$-dimentional full-rank integer lattice $\Lambda\in\mathbb{Z}^{m}$ can be defined as the set of all integer linear combinations of $m$ linearly independent over $\mathbb{R}$ basis vectors $\textbf{B}=\{\vec{b}_{1},\ldots,\vec{b}_{m}\}\subset\mathbb{Z}^{m}$: $$\lambda=\mathcal{L}(\textbf{B})=\left\{\textbf{B}\vec{c}=\sum_{i=0}^{i=m} c_{i}\vec{b}_{i}:\vec{c}\in\mathbb{Z}^{m}\right\}$$
Now, we want to consider group theoretic definition of a lattice:
$m$-dimentional full-rank integer lattices $\Lambda\in\mathbb{Z}^{m}$ are discrete additive subgroups of $\mathbb{Z}^{m}$ having finite index, i.e., the quotient group $\frac{\mathbb{Z}^{m}}{\Lambda}$ (its elements are sets) is finite.
[David Cash, Dennis Hofheinz, Eike Kiltz, Chris Peikert."Bonsai Trees, or How to Delegate a Lattice Basis", In Journal Of Cryptology 2012]
Question 1: Is there any intuitive concrete example of this quotient group modulo a lattice, $\frac{\mathbb{Z}^{m}}{\Lambda}$,?
A lattice is $\color{red}{infinite}$, but lattice crypto actually uses the $\color{red}{finite\ Abelian\ group}$ $\frac{\mathbb{Z}^{m}}{\Lambda}$. It works modulo the lattice $\Lambda$.
[Phong Nguyen. "Lattice based Cryptography Slides" In Post Quantum Cryptography Winter School 2016]
Question 2: Does $\frac{\mathbb{Z}^{m}}{\Lambda}$ being a finite group has any uses in lattice cryptography (like in hardness of some problems, or in security reductions, ...)? Or it happens to $\Lambda$ has a finite index, and it's just a mathematical property (not interesting in cryptography)?