This is most likely a silly English question, but in Roman's "Advanced Linear Algebra," on page 21, he writes that:
Let $S$ be a subset of a commutative ring $R$ with identity. Let $\equiv$ be the binary relation on $R$ defined by $a \equiv b \iff a - b \in S$. It is easy to see that $\equiv$ is an equivalence relation.
Does this mean that $S$ contains the identity of $R$? I can't see it any other way because if $\equiv$ (as defined) were an equivalence relation, then by reflexivity, $a - a = 0$ must be an element of $S$. However, the sentence to me sounds like it's saying $R$ is a ring with identity and that we don't know anything about $S$ other than it is a subset of $R$.
The book is wrong: $\equiv$ is an equivalence relation iff $S$ is a subgroup of the additive group of $R$.