Ring subset which absorbs but is not an additive subgroup

Question

Are there any undegrad-level examples of ring subsets which possess absorbtion property (as in ideal definition) but are not ideals (i.e. are not additive subgroups)?

Answer 1

The union of two ideals will possess the absorption property, but not necessarily be closed under addition and subtraction. For example, in $\mathbb Z$, we have $2\mathbb Z \cup 3\mathbb Z$, which contains $2$ and $3$ but not $1=3-2$.