Ideals in a ring with identity Let $R$ be a ring with identity element 1 and $I$ an ideal not containing 1.  Why is $I$ not a subring of $R$?
 A: By definition a subring must contain the identity element of the ring.  If $I$ does not contain $1$ then by definition $I$ is not a subring.
This doesn't mean that $I$ is not a ring.  For example $\mathbb{R \times R}$ (pairs of real numbers) is a ring.  The multiplication and addition are $(a, b) + (c, d) = (a + c, b + d)$ and $(a, b)(c, d) = (ac, bd)$.  The set of pairs $I = \{(a, 0) \ | \ a \in \mathbb R\}$ is an ideal in this ring and it is certainly a ring in its own right (it's the ring $\mathbb R$).  But it's not a subring of $\mathbb{R \times R}$ because the unit in $\mathbb{R \times R}$ is $(1, 1)$ and this element is not contained in $I$.
Another way of putting this is to say that a subset $I \subseteq R$ of a ring is a subring if $I$ is a ring and the inclusion map $I \hookrightarrow R$ is a homomorphism of rings.  The inclusion $I \hookrightarrow \mathbb{R \times R}$ respects multiplication and addition, but it does not map the identity element $(1, 0) \in I$ to the identity element $(1, 1) \in \mathbb{R \times R}$ so it is not a ring homomorphism.
A: A subring of $R$ must be a subgroup, closed under multiplication. It must contain the multiplicative identity iff your definition of a ring requires $R$ to have a multiplicative identity.
However, it is possible to have a subset of a ring with identity, $R$, that can be considered a ring in it's own right, even though it does not contain $1$ (and thus is not a subring of R).
A ring element is called "idempotent" if $r^2 = r$. It is a fact (which you might like to prove) that for any idempotent $r$ (in a ring $R$ with 1, where multiplication is commutative) that the ideal generated by $r$, $(r)$ can naturally be thought of as a ring even though it is not necessarily a subring. (Take for example $(3)$ in $\mathbb{Z_6}$.)
A: Some authors require rings to be unital (have an identity), but others don't. In fact the second allow rings to have an identity but not as part of the ring definition. In the second group all ideals are subrings. The Bourbaki group appears to favor the definition of the first group. Ring theory is quite extensive in both groups. There are situations where the first group's definition enable certain results for example if a ring has an identity (or right or left identity) , it has a maximal ideal (right and left ideal) by using Zorn's  lemma.
Hopkin's theorem "every Artinian ring with identity is Noetherian is a place where the identity is important. 
