Semigroups (ideals of a semigroup)

Question

How many ideals are there in the $\mathbb Z_{28}$?

$\mathbb Z_{28}=\{0, 1, 2, 3, 4, 5, ..., 27\}$ is a semigroup under multiplication modulo 28.

Answer 1

As a ring, $\mathbb{Z}_{28}$ is isomorphic to $\mathbb{Z}_{7} \times \mathbb{Z}_{4}$. The multiplicative structure of $\mathbb{Z}_{7}$ is a cyclic group of order $6$ plus a zero. It contains two (nonempty) ideals: $0$ and $\mathbb{Z}_{7}$. The multiplicative structure of $\mathbb{Z}_{4}$ is a semigroup with a chain of three $\mathcal{J}$-classes: a cyclic group of order $2$ ($\{1, 3\}$), a nonregular $\mathcal{J}$-class ($\{2\}$) and a zero $(\{0\})$. It contains three (nonempty) ideals: $\{0\}$, $\{0, 2\}$ and $\mathbb{Z}_{4}$. Thus the multiplicative semigroup of $\mathbb{Z}_{28}$ has $2 \times 3 = 6$ (nonempty) ideals.