Discussion
Consider the following statements :
I : Any finite non-commutative ring must have at least 16 elements.
II : Let R be a ring with unity such that |R| = 9. Then R is commutative.
Which of the following is true ?
Answer: B
Only II is true.
II: a ring with unity of order 9 has additive group \(\mathbf{Z}_{9}\) or \(\mathbf{Z}_{3} \times \mathbf{Z}_{3}\). In the first case 1 generates the additive group, so every element is an integer multiple of 1 and the ring is commutative. In the second, the subring generated by 1 is \(\mathbf{Z}_{3}\), and adding one more element a spans the ring as \(\{m + na\}\); any two such elements commute. So every ring with unity of order \(p^{2}\) is commutative.
I is false: the ring of upper triangular 2 × 2 matrices over \(\mathbf{Z}_{2}\) has only 8 elements and is not commutative, and without requiring unity there are non-commutative rings of order 4. So 16 is not a lower bound.
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