# nov07 - n elements then every quotient of M may be...

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Math 5125 Monday, November 7 November 7, Ungraded Homework Exercise 10.2.7 on page 350 Let z be a ﬁxed element of the center of R . Prove that the map m 7→ zm is an R -module homomorphism from M to itself. Show that for a commutative ring R the map from R to End R ( M ) given by r 7→ rI is a ring homomorphism (where I is the identity endomorphism). Denote by θ the map m 7→ zm . Then for m , n M and r R , we have θ ( m + n ) = z ( m + n ) = zm + zn = θ m + θ n , and because z is in the center of R , we have zr = rz and hence θ ( rm ) = zrm = rzm = r θ m . This proves that θ End R ( M ) . Now suppose that R is commutative. This means that rI End R ( M ) for all r R , because it is the map m 7→ rm for m M , so we could use the above. Let us denote by φ the map r 7→ rI . Then for r , s R , we have φ ( r + s )( m ) = ( r + s ) m = rm + sm = φ ( r ) m + φ ( s ) m = ( φ ( r )+ φ ( s ))( m ) for m M , which shows that φ ( r + s ) = φ ( r )+ φ ( s ) . Also φ ( rs )( m ) = ( rs ) m = r ( sm ) = φ ( r ) φ ( s ) m for all m M , which shows that φ ( rs ) = φ ( r ) φ ( s ) . This proves that φ is a ring homomor- phism as required. Exercise 10.3.6 on page 356 Let R be a ring with a 1. Prove that if M is a ﬁnitely generated R -module that is generated by
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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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