Discussion
Let R be a ring with identity \(1\ (\ne 0)\) and \(m\) be the characteristic of R, then which of the following is true ?
Answer: C
The true statement is wherever R is finite, \(m \ne 0\).
In a finite ring the elements \(1, 1 + 1, 1 + 1 + 1, \ldots\) cannot all be distinct, so \(k \cdot 1 = 0\) for some positive \(k\), and the characteristic is positive.
The other statements fail. \(\mathbf{Z}_{p}[x]\) is infinite with characteristic \(p\), which refutes A (infinite forces \(m = 0\)) and B (\(m \ne 0\) forces finite). \(\mathbf{Z}_{p}\) itself is finite with \(m = p \ne 0\), which refutes D (\(m \ne 0\) forces infinite).
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