Quick question, mostly just for my knowledge, but I'm working on a problem:
Determine whether the indicated set $A$ is an ideal in the indicated ring $R$:
$$A = \{0,2,4,6,8\},~~R = \mathbb{Z}/10\mathbb{Z}$$
For short hand, can I denote $A$ as $2\mathbb{Z}/10\mathbb{Z}$?
Yes. $A=2\mathbb{Z}/10\mathbb{Z} = \left\{ 2k+10\mathbb{Z}\;|\; k \in \mathbb{Z} \right\} = \left\{ 10\mathbb{Z}, 2+10\mathbb{Z}, \dots, 8+10\mathbb{Z}\right\}$.
Also, by the third isomorphism theorem: $R/A = (\mathbb{Z}/10\mathbb{Z})/(2\mathbb{Z}/10\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$ (the field of order 2).