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" ]

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given a ring R with exactly two right ideals, one of which is necessarily R itself and the other is {0}, since every ring has at least these two ideals. ‖ ‖ 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.⚹◊What is the definition of a right ideal in a ring $R$?⍭ Generate me a similar question◊

Sia