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I am currently going through Lang's Algebra, and I've come across a definition that a friend had warned me I would eventually encounter and said that Lang had defined incorrectly. He then cited the 'Terminology' section of this page: https://en.wikipedia.org/wiki/Monomorphism.

Lang's definition is as follows:

"If a homomorphism $u: N \rightarrow M$ is such that $$0 \rightarrow N \xrightarrow{u} M$$ is exact, then we also say that $u$ is a monomorphism or an embedding. Dually, if $$N \xrightarrow{u} M \rightarrow 0$$ is exact, we say that $u$ is an epimorphism."

It seems to me that this is actually the same definition as the one currently in use today in category theory. Is my friend incorrect and misremembering the source of the wrong definition, or is Lang's definition indeed incorrect?

abstract
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  • Did you really mean to assert $N$ to be "a submodule" of $M$? This precedes what you indicate to be Lang's (quoted) definition. – hardmath Sep 24 '16 at 02:19
  • Sorry, he had defined $M$ and $N$ in this way in the paragraph before the quoted definition, and I assumed (incorrectly?) that the definitions of $M$ and $N$ carried over. I have edited the post to correct this. – abstract Sep 24 '16 at 03:22

1 Answers1

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A monomorphism in a category is an arrow that is left cancellable, an epimorphism is an arrow that is right cancellable.

Lang defines monomorphisms in a category of modules to be injective maps (equivalently, those with zero kernel) and epimorphisms to be those surjective maps (equivalently, those with zero cokernel).

These definitions agree in any module category, but fail to agree in other categories.

Pedro
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