TITLE: What is the characteristic of a ring that does not contain an identity element?
QUESTION [6 upvotes]: Does a ring without an identity element even have a characteristic?

REPLY [12 votes]: One may define the characteristic of an abelian group $A$ to be the non-negative generator of the ideal $\mathrm{Ann}(A)$. Thus, we have $\mathrm{char}(A) \cdot A = 0$, and every other integer with this property is a multiple of $\mathrm{char}(A)$. This coincides with the usual definition of the characteristic of a ring when we consider its underlying additive group. As you see, we don't really need a unit, in fact no multiplication.