Unformatted text preview: n elements, then every quotient of M may be generated by n elements. Deduce that quotients of cyclic modules are cyclic. Let M be generated by the n elements x 1 , . . . , x n , and let M / N be a quotient of M , where N is a submodule of M . Thus we have M = Rx 1 + ··· + Rx n and given m ∈ M , we can ﬁnd r 1 , . . . , r n ∈ R such that m = r 1 x 1 + ··· + r n x n . Now consider M / N ; the general element can be written in the form m + N , where m ∈ M . Write m = r 1 x 1 + ··· + r n x n . Then m + N = r 1 ( x 1 + N )+ ··· + r n ( x n + N ) , which shows that M / N is generated by x 1 + N , . . . , x n + N . We deduce that M / N can be generated by n elements. The ﬁnal statement, that quotients of cyclic modules are cyclic, is just the case n = 1....
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 Fall '07
 PALinnell
 Math, Algebra, Ring, Abelian group, θ, Ring theory, Rn xn

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