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I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series)
In Chapter 1: Basics we find Corollary 1.16 on module homomorphisms and quotient modules. I need help with some aspects of the proof.
Corollary 1.16 reads as follows:
View attachment 3258
View attachment 3259
In the above text, the first line of the Proof reads as follows:
"If such a mapping $$f'$$ exists, it must satisfy
$$(x + M')f' = xf$$ ... ... ... (1.17)
and this shows that there can be at most one such mapping. ... ... "
Can someone please explain why $$f'$$ must satisfy 1.17 and, further, why there can be at most one such mapping?
Further, the next sentence of the proof reads:
"Since $$M' \subseteq ker f, xf$$ is independent of its choice in the coset ... ... "
Can someone please explain why $$xf$$ being independent of its choice in the coset, depends on $$M' \subseteq ker f$$?
Help would be appreciated.
Peter
In Chapter 1: Basics we find Corollary 1.16 on module homomorphisms and quotient modules. I need help with some aspects of the proof.
Corollary 1.16 reads as follows:
View attachment 3258
View attachment 3259
In the above text, the first line of the Proof reads as follows:
"If such a mapping $$f'$$ exists, it must satisfy
$$(x + M')f' = xf$$ ... ... ... (1.17)
and this shows that there can be at most one such mapping. ... ... "
Can someone please explain why $$f'$$ must satisfy 1.17 and, further, why there can be at most one such mapping?
Further, the next sentence of the proof reads:
"Since $$M' \subseteq ker f, xf$$ is independent of its choice in the coset ... ... "
Can someone please explain why $$xf$$ being independent of its choice in the coset, depends on $$M' \subseteq ker f$$?
Help would be appreciated.
Peter
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