1.5 Modules
7
1.5 Modules
In this subsection we discuss a little bit about vectorial modules. Roughly speaking,
a module is a “vector space-like”, where the scalars belong to a ring and not in a field
(the reader can compare Definitions 1.1.1 and 1.5.1 to see the complete similarity of
both definitions). We next give the formal definition of a module.
Definition 1.5.1 Assume that R is a commutative ring with unit. An R-module is
a set M endowed with two operations, an (internal) operation + : M × M −→ M
(called addition), and an external operation · : R × M −→ M (called scalar multiplication) such that + satisfies
(A1) ∀ v ∈ M, there exists 0 ∈ M such that v + 0 = 0 + v = v (identity);
(A2) ∀ u, v, w ∈ M, u + (v + w) = (u + v) + w (associative);
(A3) ∀ v ∈ M, there exists −v ∈ M such that v + (−v) = (−v) + v = 0 (symmetric);
(A4) ∀ u, v ∈ M, u + v = v + u (commutative);
and the operations · and + satisfy
(M1) ∀ u, v ∈ M and ∀ r ∈ R, r (u + v) = ru + r v;
(M2) ∀ u ∈ M and ∀ r 1 , r 2 ∈ R, (r 1 + r 2 )u = r 1 u + r 2 u;
(M3) ∀ u ∈ M and ∀ r 1 , r 2 ∈ R, (r 1 r 2 )u = r 1 (r 2 u);
(M4) ∀ u ∈ M and 1 ∈ R, 1m = m.
Remark 1.5.1 Note that an R-module is, in fact, an ordered triple of the form
(M, +, ·). However, to avoid stress of notation, we simply write M when the operations + and · are known from the context.
The concepts of submodule and homomorphism of modules are also analogous
to that of subspace and linear transformation, respectively, as the reader can see in
the following definition.
Definition 1.5.2 Let R be a commutative ring with unit and let M be an R-module.
A subset N ⊆ M is a called an R-submodule of M if N is an additive subgroup of
M closed under scalar multiplication.
Definition 1.5.3 Let R be a commutative ring with unit and let M and N be two
R-modules. A function h : M −→ N is called an R-homomorphism if
(H1) ∀ u, v ∈ M, h(u + v) = h(u) + h(v);
(H2) ∀ r ∈ R and ∀ u ∈ M, h(ru) = rh(u).
A special class of module is the class of free-modules, i.e., modules that have
basis.
7
1.5 Modules
In this subsection we discuss a little bit about vectorial modules. Roughly speaking,
a module is a “vector space-like”, where the scalars belong to a ring and not in a field
(the reader can compare Definitions 1.1.1 and 1.5.1 to see the complete similarity of
both definitions). We next give the formal definition of a module.
Definition 1.5.1 Assume that R is a commutative ring with unit. An R-module is
a set M endowed with two operations, an (internal) operation + : M × M −→ M
(called addition), and an external operation · : R × M −→ M (called scalar multiplication) such that + satisfies
(A1) ∀ v ∈ M, there exists 0 ∈ M such that v + 0 = 0 + v = v (identity);
(A2) ∀ u, v, w ∈ M, u + (v + w) = (u + v) + w (associative);
(A3) ∀ v ∈ M, there exists −v ∈ M such that v + (−v) = (−v) + v = 0 (symmetric);
(A4) ∀ u, v ∈ M, u + v = v + u (commutative);
and the operations · and + satisfy
(M1) ∀ u, v ∈ M and ∀ r ∈ R, r (u + v) = ru + r v;
(M2) ∀ u ∈ M and ∀ r 1 , r 2 ∈ R, (r 1 + r 2 )u = r 1 u + r 2 u;
(M3) ∀ u ∈ M and ∀ r 1 , r 2 ∈ R, (r 1 r 2 )u = r 1 (r 2 u);
(M4) ∀ u ∈ M and 1 ∈ R, 1m = m.
Remark 1.5.1 Note that an R-module is, in fact, an ordered triple of the form
(M, +, ·). However, to avoid stress of notation, we simply write M when the operations + and · are known from the context.
The concepts of submodule and homomorphism of modules are also analogous
to that of subspace and linear transformation, respectively, as the reader can see in
the following definition.
Definition 1.5.2 Let R be a commutative ring with unit and let M be an R-module.
A subset N ⊆ M is a called an R-submodule of M if N is an additive subgroup of
M closed under scalar multiplication.
Definition 1.5.3 Let R be a commutative ring with unit and let M and N be two
R-modules. A function h : M −→ N is called an R-homomorphism if
(H1) ∀ u, v ∈ M, h(u + v) = h(u) + h(v);
(H2) ∀ r ∈ R and ∀ u ∈ M, h(ru) = rh(u).
A special class of module is the class of free-modules, i.e., modules that have
basis.