show that the frist ring is not isomorphic to the second a. R x R x R x R and M(R) where the R\'s are reals b. Q and R where Q are rationals and R are reals c. Z4 x Z4 and Z16 Solution Try comparing orders. Since Z16 is cyclic, it has an element of order 16. On the other hand, Z4 x Z4 has no elements of order 16, because the highest order it has are elements of order 4 (arising from one or both of the copies of Z4). Hence, these groups can\'t be isomorphic. .

1. show that the frist ring is not isomorphic to the second
a. R x R x R x R and M(R) where the R's are reals
b. Q and R where Q are rationals and R are reals
c. Z4 x Z4 and Z16
Solution
Try comparing orders.
Since Z16 is cyclic, it has an element of order 16.
On the other hand, Z4 x Z4 has no elements of order 16, because the highest order it has are
elements of order 4 (arising from one or both of the copies of Z4).
Hence, these groups can't be isomorphic.