TITLE: Does every unital ring contain all the integers?
QUESTION [26 upvotes]: Let us suppose there is a ring $R$ with the multiplicative identity $1$. 

We know that $1+r\in R$, where $r$ is any element of the ring $R$. Does this mean $1+1$ is also part of the ring, or does $r$ have to be an element of the ring different from $1$?
Is $1+1$ called $2$ in the ring? Similarly, as $-1$ is also part of the ring, is $-1+ -1$ called $-2$ in the ring? If it is, then I suppose all integers are contained in every unital ring. 

These questions are very elementary. However, I read contradictory remarks in some places which tend to confuse me. So I thought it would be best to clear any doubts, however trivial the questions.
Thanks in advance for your help!

REPLY [3 votes]: The theory of rings has many constants. While $0$ and $1$ are the only constants introduced by the usual definition of a ring, the operations $+$, $-$, and $\cdot$ allow us to construct more constants. It is useful to use "2" to refer to the constant $1+1$, "3" for $2+1$, and so forth.
Incidentally, we can also construct additional operations; e.g. "multiplication by 3", or "raising to the seventh power" are also ring operations. In fact, for every polynomial $f$ in zero or more variables with integer coefficients, or even "plug values into $f$" is also a ring operation. (and every ring operation -- including the constants -- is of this form)

Another convention that is often useful is that when we have a ring homomorphism $\varphi :R \to S$, it can be useful to represent elements of $S$ by specifying elements of $R$. e.g. if $r \in R$, we might also use the notation $r$ for the element $\varphi(r) \in S$ in any setting where it's clear we are talking about elements of $S$.
Some situations where this convention is particularly useful are

$\varphi$ is monic, and we want to think of $R$ as a subring of $S$
$\varphi$ is surjective, and we want to think of $S$ as a quotient of $R$ (i.e. $S$ is the ring $R$ modulo an equivalence relation)
There is only one map $R \to S$, such as when $R = \mathbb{Z}$.

Note that the third bullet above results in the exact same notation as suggested in the previous section. (which is part of the reason why it's useful notation!)