(i) Given a pushout diagram in R Mod: prove that g injective implies αinjective and that g surj

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(i) Given a pushout diagram in R Mod: prove that g...

(i) Given a pushout diagram in ${ }_{R}$ Mod:
prove that $g$ injective implies $\alpha$ injective and that $g$ surjective implies $\alpha$ surjective. Thus, parallel arrows have the same properties.

Key Concepts

Pushout

A pushout is a colimit in category theory that 'glues' two objects together along a common subobject. In the context of module theory, it represents an object together with two maps from two modules such that any pair of maps from these modules that agree on the common part factors uniquely through the pushout. This construction is a key tool in proving properties such as the preservation of injectivity or surjectivity between corresponding arrows in the diagram.

Universal Property

The universal property of the pushout ensures that any commutative diagram involving the two maps from the original objects can be uniquely factored through the pushout. This property is crucial because it enforces that structural properties of morphisms, like being injective or surjective, are transferred between the parallel arrows, as any conflicting behavior would violate the uniqueness guaranteed by the pushout.

Injective and Surjective Morphisms

In module theory, an injective morphism is one that preserves distinctness (i.e., no two different elements are mapped to the same element) and a surjective morphism is one that covers the entire target module. These properties are fundamental in understanding module homomorphisms. In the context of a pushout diagram, these properties on one arrow translate to the other parallel arrow due to the way the pushout is constructed, preserving the overall structure in the category.

Colimits

Colimits are general constructions in category theory that encompass notions like coproducts, coequalizers, and pushouts. Understanding colimits is essential because pushouts are a specific example of colimits. They provide a framework for constructing new objects that amalgamate existing ones while preserving the mappings and properties, such as injectivity or surjectivity, across the involved morphisms.

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