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I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Section 3.2 on exact sequences in Mod_R and need help with the proof of Proposition 3.2.7.
Proposition 3.2.7 and its proof read as follows:View attachment 3612I am having trouble in understanding the proof that condition (2) implies condition (3).
Bland's argument of $$ (2) \Longrightarrow (3) $$ begins, of course, with the assumption that $$\text{ Ker } g$$ is a direct summand of $$M$$; and then Bland let's $$N$$ be a submodule of $$M$$ such that
$$M = \text{ Ker } g \ \oplus \ N $$
Given this and given that the sequence being considered is exact, we have
$$ M/ \text{ Ker } g \ \cong \ g(M) $$ by the First Isomorphism Theorem for R-modules.
Thus ... ...
$$M/ \text{ Ker } g \ \cong \ M_2$$
since $$g$$ is an epimorphism ... ...
But as you can see in the above text, Bland states that
$$ M_2 \ \cong \ M/ \text{ Ker } g \ \cong \ N $$ BUT ... ...
How does Bland deduce $$ M/ \text{ Ker } g \ \cong \ N $$?
That is why is $$ M/ \text{ Ker } g \ \cong \ N $$?
Peter
I am trying to understand Section 3.2 on exact sequences in Mod_R and need help with the proof of Proposition 3.2.7.
Proposition 3.2.7 and its proof read as follows:View attachment 3612I am having trouble in understanding the proof that condition (2) implies condition (3).
Bland's argument of $$ (2) \Longrightarrow (3) $$ begins, of course, with the assumption that $$\text{ Ker } g$$ is a direct summand of $$M$$; and then Bland let's $$N$$ be a submodule of $$M$$ such that
$$M = \text{ Ker } g \ \oplus \ N $$
Given this and given that the sequence being considered is exact, we have
$$ M/ \text{ Ker } g \ \cong \ g(M) $$ by the First Isomorphism Theorem for R-modules.
Thus ... ...
$$M/ \text{ Ker } g \ \cong \ M_2$$
since $$g$$ is an epimorphism ... ...
But as you can see in the above text, Bland states that
$$ M_2 \ \cong \ M/ \text{ Ker } g \ \cong \ N $$ BUT ... ...
How does Bland deduce $$ M/ \text{ Ker } g \ \cong \ N $$?
That is why is $$ M/ \text{ Ker } g \ \cong \ N $$?
Peter