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Definition

Thoughout let RR be some ring. Write RRMod for the category of module over RR. Write URModU R Mod \to Set for the forgetful functor that sends a module to its underlying set.

Definition

For NRModN \in R Mod a module, a submodule of NN is a subset of U(N)U(N) which

  1. is a subgroup of the underlying group (closed under the addition in NN);

  2. is preserved by the RR-action.

Equivalently this means:

Definition

A submodule of NRModN \in R Mod is a module homomorphism i:KNi : K \to N whose underlying map U(i)U(i) of sets is an injection.

And since the injections in RRMod are precisely the monomorphisms, this means that equivalently

Definition

A submodule of NRModN \in R Mod is a monomorphism i:KNi : K \hookrightarrow N in RRMod. Hence a submodule is a subobject in RRMod.

Remark

Given a submodule KNK \hookrightarrow N, the quotient module NK\frac{N}{K} is the quotient group of the underlying abelian groups.

Examples

Example

For N=RN = R regarded as a module over itself, a submodule is precisely an ideal of RR.

Example

For f:N 1N 2f : N_1 \to N_2 a homomorphism of modules,

  1. the kernel ker(f)N 1ker(f) \hookrightarrow N_1 is a submodule of N 1N_1,

  2. the image im(f)N 2im(f) \hookrightarrow N_2 is a submodule of N 2N_2.

Remark

In example 2 quotient module of N 2N_2 by the image is the cokernel of ff

coker(f)N 2im(f). coker(f) \simeq \frac{N_2}{im(f)} \,.

Properties

Of free modules

Let RR be a ring.

Proposition

Assuming the axiom of choice, the following are equivalent

  1. every submodule of a free module over RR is itself free;

  2. every ideal in RR is a free RR-module;

  3. RR is a principal ideal domain.

A proof is in (Rotman, pages 650-651).

References

For instance

  • Rotman Advanced Modern Algebra

Revised on September 24, 2012 23:28:02 by Urs Schreiber (82.169.65.155)