Question

An element of R is called idempotent if a ²= a. find all idempotents of Z₆ x Z₁₂

Answer 1

$\text{Since the idempotent in $\Z_6$ are $\{0, 1, 3,4\}$ and those in $\Z_{12}$ are \{0,1,4,9\}} \\ \text{Then we have that the idempotent in $\Z_6 \times \Z_{12}$ is the set of the cartesian product of the idempotent in each of the constituent sets.} \\ \therefore \text{Idempotent}(\Z_6, \Z_{12}) = \{ (0,0), (0,1), (0,4), (0,9),(1,0),(1,1),(1,4),(1,9),(3,0), (3,1), (3,4), (3,9), (4,0), (4,1),(4,4),(4,9)\}$