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## Main Question or Discussion Point

Let ##M## be a left R-module and ##f:M \to M## an R-endomorphism.

Consider this infinite descending sequence of submodules of ##M##

##M \supseteq f(M) \supseteq f^2(M) \supseteq f^3(M) \supseteq \cdots (1)##

Can anybody show that the sequence (1) is strictly descending if ##f## is injective and ##f## is NOT surjective ?

So you have to prove that ## f^n(M) \neq f^{n+1}(M)## for all ##n > 0##, given that ##f## is injective and ##f## is not surjective.

Of course ##M \neq f(M)##.

Can someone help me with this?

Consider this infinite descending sequence of submodules of ##M##

##M \supseteq f(M) \supseteq f^2(M) \supseteq f^3(M) \supseteq \cdots (1)##

Can anybody show that the sequence (1) is strictly descending if ##f## is injective and ##f## is NOT surjective ?

So you have to prove that ## f^n(M) \neq f^{n+1}(M)## for all ##n > 0##, given that ##f## is injective and ##f## is not surjective.

Of course ##M \neq f(M)##.

Can someone help me with this?