Clarification about the definition of chracteristic of a ring 
Definition: let $(R,+ \cdot)$ we define the  characteristic of a ring $R$ like the least positive integer  $n$ such that $a\cdot n=0 \forall a\in R$  and we denote it like a $char(R)=n$.In case of that integer does not Exist then we say that the ring $R$ have characteristic $0$

Now if we have that the Ring  that have a $1$, then we can think the $char(R)$ like the order of the element $1$ in $(R,+)$, but in otherwise if the ring does not contains $1$ then we should can use the other definition and find an example of a ring $(R,+ \cdot)$ wihout identinty $1$ that have an other characteristic non zero.
Or is true my intuition that say "If we have a ring without $1$ then $char(R)=0$
Can someone give me an contra example or proof about my intuition.
 A: No, it is not true that the characteristic will be zero. Both ways can happen.
Let $F_p$ denote the field of $p$ elements where $p$ is a prime.
The subset of finitely nonzero elements of $\prod_{i\in\mathbb N}F_2$ (you can write it $\bigoplus_{i\in\mathbb N}F_2$) is a ring without identity and of characteristic $2$.
Then if you enumerate the primes as $p_i$, the subset of finitely nonzero elements of $\prod_{i\in\mathbb N}F_{p_i}$ (you can write it $\bigoplus_{i\in\mathbb N}F_{p_i}$) is a ring without identity and of characteristic $0$.
Addition and multiplication in both rings is coordinatewise, of course, and neither one has an identity.
A: The ring of even integers modulo $8$, $\{\overline{0},\overline{2},\overline{4},\overline{6}\}$ with the usual addition and multiplication modulo $8$, is a ring without $1$; the characteristic is $4$.
More generally, for any ring $R$ and $a\in R$, there is a group homomorphism $\mathbb{Z}\to R$ given by $1\longmapsto a$. This group homomorphism has a kernel, which being a subgroup of $\mathbb{Z}$ must be of the form $(n_a)$ for some $n_a\geq 0$. This is called the "annihilator of $a$ in $\mathbb{Z}$".
The intersection of all these subgroups is also a subgroup, $\bigcap_{a\in R}(n_a)$, the "annihilator of $R$ in $\mathbb{Z}$" (we are vieweing $R$ as a $\mathbb{Z}$-module here). This subgroup is of the form $(n_R)$ for some $n_R\geq 0$; that $n_R$ is the characteristic of $R$.
If $b=ar$ or $b=sa$ (that is, if $a$ divides $b$ on the left or on the right), then $(n_a)\subseteq (n_b)$. Indeed, if $ka=0$, then $kb=k(ar)=(ka)r = 0r = 0$ or $kb=k(sa) = s(ka) = s0 = 0$. Thus, in particular, if $u$ is either left invertible or right invertible, then $(n_u)\subseteq (n_a)$ for any $a\in R$, so you can find the characteristic by considering only $u$. In particular, by considering $1$ when it exists. But in general, you may need to do more work.
For another example, consider the ring:
$$R=\bigoplus_{n=1}^{\infty}\mathbb{Z}/n\mathbb{Z}.$$
This is a ring without unity (since every element has finite support), and even though each $x$ has the property that $(n_x)\neq (0)$, nonetheless the characteristic is $0$ because for every $k\in\mathbb{Z}$ you can find an element $y$ with $ky\neq 0$. In fact, no finite collection of elements will determine the characteristic of $R$, you need to consider infinitely many elements.
