symmetric monoidal (∞,1)-category of spectra
Thoughout let $R$ be some ring. Write $R$Mod for the category of module over $R$. Write $U R Mod \to$ Set for the forgetful functor that sends a module to its underlying set.
For $N \in R Mod$ a module, a submodule of $N$ is a subset of $U(N)$ which
Equivalently this means:
A submodule of $N \in R Mod$ is a module homomorphism $i : K \to N$ whose underlying map $U(i)$ of sets is an injection.
And since the injections in $R$Mod are precisely the monomorphisms, this means that equivalently
A submodule of $N \in R Mod$ is a monomorphism $i : K \hookrightarrow N$ in $R$Mod. Hence a submodule is a subobject in $R$Mod.
Given a submodule $K \hookrightarrow N$, the quotient module $\frac{N}{K}$ is the quotient group of the underlying abelian groups.
For $f : N_1 \to N_2$ a homomorphism of modules,
In example quotient module of $N_2$ by the image is the cokernel of $f$
Let $R$ be a ring.
Assuming the axiom of choice, the following are equivalent
every submodule of a free module over $R$ is itself free;
every ideal in $R$ is a free $R$-module;
$R$ is a principal ideal domain.
A proof is in (Rotman, pages 650-651).
For instance
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