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Which of the following statements are true? Give reasons for your answers, in the form of a short proof or a counterexample. i) M3 (Z) has no nilpotent elements. ii) If P and 1 P are prime ideals of a ring 2 ,R then . P1 P2 = P1 ∩ P2 iii) The set of cosets of <(1 2)> in 3S is a group with respect to multiplication of cosets. iv) If (G,*) is a group, then , * is the only binary operation defined on .G v) If every element of a group G has finite order, then G must be of finite order. vi) If k is a field, so is k × k. vii) x is a unit in R[x]. viii) If A and B are two sets such that A∪B = fi then A ∩B =fi. ix) Q[x]/<x^6+17> is a field of characteristic 6. x) Any two groups of order m are isomorphic, where m∈N.
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