66. Let R be a ring with multiplicative identity. If U is an additive subgroup of R such that ur "belongs to" U for all u "in" U and for all r "in" R, then U is said to be a right ideal of "R". If R has exactly two right ideals, which of the following must be true?

I. R is commutative

II. R is a division ring (that is all elements except the additive identity have inverses)

III. R is infinite

I tried it the following way (Comments are welcome).

Since the question talks of right ideals, we cannot conclusively say that R is commutative.

Since R has exactly two right ideals, they must be {0} (additive identity) and R itself. Using #5 in

http://en.wikipedia.org/wiki/Ideal_(ring_theory)#Examples

we can conclude that R is a division ring or a field.

Since the commutativity of R cannot be established, it is a division ring.

Finally the division ring of quaternions is proof enough that R need not be infinite.

Hence only II is acceptable (I wonder how I would be able to solve a similar question in an exam with 2.5 min/question on the average)!