Unformatted text preview: R S D h S i C R S . Proof. It is straightforward to check that R S is a left ideal. h S i C R S is a sum of subgroups R , so it is a subgroup. Moreover, for r 2 R , we have r h S i ± R S . It follows from this that h S i C R S is a left ideal. If J is any left ideal of R containing S , then J ´ h S i , because J is a subgroup of R . Since J is a left ideal, J ´ R S as well. Therefore J ´ h S i C R S . This shows that h S i C R S is the smallest left ideal containing S . The intersection of all left ideals of R containing S is also the smallest left ideal of R containing S , so (b) follows. Finally, if R has an identity element, then S ± R S , so h S i ± R S , which implies (c). n...
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 Fall '08
 EVERAGE
 Algebra, Normal subgroup, Ring, Empty set, rs

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