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Question
Math
Posted 8 months ago
Let R be a ring with a multiplicative identity. If U is an additive subgroup of R such that ur in 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 multiplicative inverses). III. R is infinite.[ "I only", "II only", "III only", "I and II only" ]
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Answer from Sia
Posted 8 months ago
Solution by Steps
step 2
Since there are no other right ideals, every non-zero element of R must generate R as a right ideal when multiplied by any element of R
step 3
This implies that every non-zero element has a multiplicative inverse, making R a division ring
step 4
The fact that R is a division ring does not necessarily imply that R is commutative, so R being commutative (I) is not guaranteed
step 5
There is no information about the size of R, so we cannot conclude that R is infinite (III)
B
Key Concept
Division Ring
Explanation
A ring with exactly two right ideals must be a division ring because every non-zero element must have a multiplicative inverse. However, this does not imply that the ring is commutative or infinite.

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