Answer to Question #47264 in Abstract Algebra for ayush kumar

Question

Which of the following statements are true ? Give reasons for your answers.
i) For any set S, S * S is an equivalence relation.
ii) If H equal to G such that IG:HI = 3, then H is normal in G.
iii) The group of units of ,
13Z
Z with respect to multiplication, is cyclic.
iv) 20 Z is an abelian group with no proper non-trivial normal subgroups (i.e., it is simple).
v) A group of order 168 has either 1 or 8 elements of order 7.
vi) In Z100 there is a unique zero divisor which is also a unit.
vii)
a, b, c R
c 0
a b
is a ring with respect to the usual matrix addition and
multiplication.
viii) Every subring of a commutative ring R is an ideal of R.
ix) Any abelian group is a ring with respect to a suitably defined multiplication.
x) If I and J are ideals in a ring R, then IJ = IJ. (