Let E be an essential extension of a left R-module M. If φ: E →N is an R-map with φ|M injective,

step 1 Hello there. In this exercise we need to state if this transformation is an isomorphism. In this

Let E be an essential extension of a left R-module M. If φ:...

Let $E$ be an essential extension of a left $R$-module $M$. If $\varphi: E \rightarrow N$ is an $R$-map with $\varphi \mid M$ injective, prove that $\varphi$ is injective.
Hint. Consider $M \cap \operatorname{ker} \varphi$.

Key Concepts

Essential Extension

An essential extension of a module is an extension in which the original module is 'large' inside the extending module: every nonzero submodule of the larger module has a nonzero intersection with the original module. This concept is central to understanding how properties such as injectivity can extend from a submodule to the whole module, since the original module touches every part of the extension.

Module Homomorphism

A module homomorphism (or R-map) is a function between modules that preserves the module operations: it is additive and respects scalar multiplication. Such maps are the fundamental morphisms in module theory and play a key role in transferring properties between modules, such as the property of being injective.

Kernel of a Module Homomorphism

The kernel of a module homomorphism is the set of elements in the domain that are mapped to the zero element in the codomain. It is a submodule of the domain and measures the failure of a homomorphism to be injective; a trivial kernel indicates that the map is injective.

Injectivity

An injective (or one-to-one) map is a homomorphism that maps distinct elements of the domain to distinct elements of the codomain, which equivalently means that its kernel contains only the zero element. In the context of essential extensions, proving that a map is injective often involves demonstrating that any nonzero element in the kernel would contradict the injectivity on the essential submodule.

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