My book defines an ideal to be a subring of R such that $xr \in I$ and $rx \in I$, whenever $r \in R$ and $x \in I$. However, by definition any subring of a ring with unity must contain unity. So it follows that in any ring with unity, the only ideal is itself.
Surely my reasoning is incorrect, but I can't figure out where I'm going wrong.
Best Answer
If your book is defining an ideal to be a subring of $R$ with the multiplication property you described above, then it is almost certain that a subring of a ring with unity need not contain unity, because as you pointed out, this would imply that the only ideal of a unital ring is the entire ring. This definition would cause us a lot of trouble for many reasons: for example, the statement that every ideal is the kernel of some ring homomorphism and vice versa would fail massively. To resolve this dilemma, you could say that an ideal is simply an additive subgroup of the ring with the "absorption" multiplication property you described above, or you could allow unital rings to have nonunital subrings (e.g. $n\mathbb{Z}\subseteq\mathbb{Z}$ for $n\neq\pm 1$).