An exercise from Artin's Algebra:
Let S be a set with a law of composition: A partition $\Pi_1 \cup \Pi_2 \cup ...$ of S is compatible with the law of composition if for all i and j, the product set
$$\Pi_i \Pi_j = \{xy \mid x \in \Pi_i, y \in \Pi_j\}$$
is contained in a single subset $\Pi_k$ of the partition.
(a) The set $\mathbb Z$ of integers can be partitioned into the three sets [Pos], [Neg], [{O}]. Discuss the extent to which the laws of composition $+$ and $\times$ are compatible with this partition. (b) Describe all partitions of the integers that are compatible with the operation $+$.
A) Yes for $\times$. No for $+$ because $4+-2=2 \in$ [Pos] but $5+-7=-2 \in$ [Neg]
B) This is an introduction to the next section on Modular Arithmetic. Some partitions are $\{\mathbb Z/\mathbb Zn\}_{n \ge 0}$. As Arturo Magidin points out, other partitions are $\{ \{a + \mathbb Zn\}_{a \in \mathbb Z} \}_{n \ge 0}$ (I'm actually not sure about the notation)
Am I correct?