Answer to Question #104947 in Abstract Algebra for Sourav Mondal

Question

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.

Answer 1

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