Question

Math

Posted 4 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 4 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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