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Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel
Statement
For an -module homomorphism , both and are submodules. Moreover,
Facts & Assumptions
Given: A homomorphism of left -modules.
A module homomorphism preserves addition and scalar multiplication, and the displayed definitions give its kernel and image (Module homomorphism and isomorphism, kernel, image and cokernel).
A nonempty subset of a module is a submodule exactly when it is closed under (The one-step submodule criterion; intersections and sums of submodules are submodules).
The additive underlying function of a module homomorphism is a group homomorphism and therefore sends to ; moreover (Module homomorphism and isomorphism, kernel, image and cokernel, A group homomorphism automatically satisfies and , and for every ; for monoid homomorphisms preservation of the identity must be assumed, In a module, , , and ).
A group homomorphism is injective exactly when its kernel is trivial (A group homomorphism is injective if and only if its kernel is trivial).
Proof
The kernel contains because . If and , then , so .
The image contains . If and , then lies in the image.
The additive underlying function has the same kernel described in [L1], so the group-homomorphism theorem gives injective exactly when .
The submodule criterion applied to steps 1.1 and 1.2 proves that and are submodules.
Together, steps 1.3 and 2.1 prove both assertions.
Depends on
- Module homomorphism and isomorphism, kernel, image and cokernel
- The one-step submodule criterion; intersections and sums of submodules are submodules
- In a module, $0_Rm=0_M$, $r0_M=0_M$, $(-r)m=-(rm)$ and $r(-m)=-(rm)$
- A group homomorphism automatically satisfies $f(e) = e'$ and $f(g^{-1}) = f(g)^{-1}$, and $f(g^{n}) = f(g)^{n}$ for every $n \in \mathbb{Z}$; for monoid homomorphisms preservation of the identity must be assumed
- A group homomorphism is injective if and only if its kernel is trivial
Used by
- Exact sequences and short exact sequences of modules Definition
- The endpoints of a short exact sequence encode injectivity and surjectivity Lemma
- First isomorphism theorem for modules: M/ker f congimf Theorem
Cited to discharge well-definedness by Module homomorphism and isomorphism, kernel, image and cokernel.
Dependency tree · two levels
27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)