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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