求助！！一道math sub题

lin2003332UID: 2661641

ets 寄得practice book 的最后一道题~
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 that belong to U and for all r that belong to R, then U is said to be a right ideal of R. If R has eaxctly two right ideals. Which of the following must be true?
1 Ris commutative
2 R is a division ring
3 R is infinite

答案是2

麻烦请高人详细讲一讲，抽象代数我没学过，一碰到这方面的题就晕~愁死了~

soarerzUID: 2721318

If you have seen some commutative rings before, then you will know that commutative ring with two ideals <=> it is a field.

2 is true:
In the general situation, notice that there are always two ideals. ( {0} and R) So these are all the possible ideals.aR = {ar | r in R} is obviously an ideal as well, and it is not zero if a is not zero, so it must be R. That means every nonzero element has an inverse. This means it is a division ring.

1 is false: consider quarternions

3 is false: consider Z/pZ, the remainders of Z dividing by p.