Let \(R\) be a ring. If there is an integer \(n\geq 1\) with \(\sum_{j=1}^n 1_R = 0_R\), the characteristic of \(R\) is the smallest such \(n\); otherwise the characteristic of \(R\) is \(0\).
The characteristic of \(R\) is the nonnegative generator \(n\) of the kernel \(n\ZZ\) of the unique ring homomorphism \(\ZZ\to R\): an integer \(m\geq 1\) lies in the kernel exactly when \(\sum_{j=1}^m 1_R=0_R\), and the kernel is \(0\) exactly when no such \(m\) exists.
What is the characteristic of a ring?