Question

. Let M be an R-module; if me M let 1(m) = {x € R |...

. Let M be an R-module; if me M let 1(m) = {x € R | xm = 0}. Show that 1(m) is a left-ideal of R. It is called the order of m. 17. If 2 is a left-ideal of R and if M is an R-module, show that for me M, λm {xm | * € 1} is a submodule of M.

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Answer #1

If then ; thus, for every .

Note that . If then (taking in the above) we get so that ; also, if then implies . Thus, is a subgroup. Also, if and then implies . Hence, is an ideal.

17. Suppose is an ideal, and . Consider .

If then for some . Thus, for every , one has since .

Note that so that . If then (taking in the above) we get ; also, if then . Thus, is a subgroup. Also, if and then . Hence, is a submodule.

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