Here is the definition of the characteristic of a ring $R$ that is common in everyday usage (for example in Lang's Algebra and Wikipedia): take the unique homomorphism $$ \mathbb{Z} \to R $$ and define $\operatorname{char}(R)$ to be the smallest nonnegative integer that generates the kernel of this map.
And yet on page A.V.2 of Bourbaki's Algebra II: Chapters 4 - 7 the characteristic is defined in a way that excludes rings without subrings which are fields. This leaves the ring of integers without a characteristic. While I can appreciate that $\mathbb{Z}$ is best considered as a mixed characteristic ring, what was the purpose of defining the characteristic like that?
I can't speak to their motivation, but it seems that a little later on they remark that a quotient of a ring has the same characteristic as the original ring, which would be false if they allowed rings not containing fields.