Can the characteristic of some ring differ from number to number? The characteristic of ring is defined as the minimum times multiplicative identity must be added to get additive identity. 
My question is, for every number of ring, can the minimum times a number has to be added to get multiplicative identity differ? (e.g. 2 needs to be added 5 times to get additive identity, while 3 needs 7 times and so forth.)
And what about the case of commutative ring?
And is there any name for number of times a number has to be multiplied to get additive identity?
 A: In the ring $\mathbb{Z}_4 = \mathbb{Z}/4\mathbb{Z}$ the characteristic is $4$, but $2 + 2 = 0$. You could call this the order of an element in the additive group, perhaps "additive order" would be a good term.
A: Given a ring $R$, if 
$$\underbrace{1_R+\cdots+1_R}_{n\text{ times}}=0_R,$$
then for any $a\in R$, we have (by the distributive property) that
$$\underbrace{a+\cdots+a}_{n\text{ times}}=a\cdot\underbrace{\left(1_R+\cdots+1_R\right)}_{n\text{ times}}=a\cdot 0_R=0_R,$$
so whatever the number is for the multiplicative identity, that is an upper bound on the number for any other element.
In the ring $R=(\mathbb{Z}/n_1\mathbb{Z})\times\cdots\times (\mathbb{Z}/n_k\mathbb{Z})$, the element
$$e_i=(\overline{0},\ldots,\underset{\substack{i\text{th}\\\text{place}}}{\overline{1}},\ldots,\overline{0})$$
has the property that
$$\underbrace{e_i+\cdots+e_i}_{n_i\text{ times}}=(\overline{0},\ldots,\overline{0})=0_R,$$
so you can pretty much make a ring with any collection of such numbers you want.

Also, I second spin's suggestion that this be referred to as the "additive order" of an element.
A: Other answers have already clarified the definition of the characteristic of a ring. But I may add one definition that is completely element-free, so hopefully this would make things clearer that characteristics do not depend on the number.
$\mathbb{Z}$ is initial in the category $\operatorname{Ring}$ (this simply means that for any ring $R$, there is a unique ring homomorphism $\mathbb{Z}\xrightarrow{\phi_R} R$),  thus a ring $R$ uniquely defines a subring of $\mathbb{Z}$, namely, $\operatorname{ker}(\phi_R)$. 
This is a subgroup of $(\mathbb{Z},+)$, hence it is generated by some $d\in\mathbb{Z}$, and we can define $d$ to be the characteristic of $R$.
You can easily show that this definition is equivalent to yours, but in this definition we have not mention any special element, thus characteristics is a property of the rings themselves.
