Topics in algebra Chapter IV: Commutative rings and modules I - 1

Example 4.9 Let M be an R-module and I be an ideal of R; then the set
n
IM = { aixi | ai ∈ I, xi ∈ M} is an R-module.
i=1
Definition 4.10 Let M and N be two R-modules. A function f : M −→ N
is a homomorphism of R-modules (R-homomorphism or R-linear map) if for
all x, y ∈ M, r ∈ R:
f(x + y) = f(x) + f(y) and f(rx) = rf(x).
f is called a monomorphism (resp. epimorphism, isomorphism) if f is injective
(resp. surjective, bijective). The kernel of f, denoted by kerf, is
the set {x ∈ M | f(x) = 0}. The image of f, denoted by Imf, is the set
{f(x) | x ∈ M}.
Notice that if f : M −→ N is an R-module homomorphism, then f is a
monomorphism if and only if kerf = 0.
In the sequel, we use the symbol ∼ = for R-module isomorphisms.
Definition 4.11 Let R be a ring and M be an R-module. A subgroup N of
M is a submodule of M if rx ∈ N whenever r ∈ R and x ∈ N.
Definition 4.12 Let N ⊆ M be R-modules; then the abelian group M/N =
{x + N | x ∈ M} is an R-module with r(x + N) = rx + N for all x ∈ M and
r ∈ R.
Example 4.13 Let f : nZ −→ mZ given by f(na) = ma is a Z-module
isomorphism. Hence nZ ∼ = mZ as Z-module.
Example 4.14 Let R be a ring; then every ideal of R is a sub-module of R.
Example 4.15 Let f : M −→ N be an R-module homomorphism; then
kerf is a submodule of M and Imf is a submodule of N.
Example 4.16 Let M be an R-module and I be an ideal of R, then IM is
a submodule of M. Moreover, if x ∈ M, then xR is a submodule of M.
Example 4.17 Let {Mi | i ∈ Λ} be R-modules; then ⊕i∈ΛMi, the direct sum
of the abelian groups Mi, is a submodule of
Mi.
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i∈Λ