Question

1.a) Obtain 3 distinct cosets of H = { I, (1 2) (3 4), (1 4) (2 3}, (1 3) (2 4)} in S4. Justify your answer.

b) Prove that if R is a finite commutative ring with identities, then every prime ideal of R is a maximal ideal
.
c) Check if the following are ring homomorphisms :

i) f: M2 (Z) -> Z : f ( a b ) = a .
( c d )
ii) f: {( a b ) I a b d belongs to Z} -> Z*Z :
0 d

f:( a b ) = ( a, d )
0 d