4.1 Submodules
Definition 4.1.1. Let \(M\) be a left \(R-\)module and \(N\) a subgroup of \(M\). Then \(N\) is a submodule (or \(R-\)
submodule) if for any \(n\in \mathbb {N}\) and any \(r\in R,\hspace {0.3cm} rn\in N\).
Note: Given a module \(M\), the set of all submodules of \(M\) forms a lattice that satisfies the modular
law.
Modular Law
Given submodules \(U, \, N_1,\, N_2\) of \(M\) such that \(N_1\subset N_2\), then \(\, \big (N_1 + U\big ) \cap N_2 = N_1 + \big (U \cap N_2\big ) \).
Definition 4.1.2. Let \(M\) and \(N\) be modules the map \(f: M\longrightarrow N\) is a homomorphism of modules if for any \(m,\, n \in M\) and \(r,\, s\in R,\) \[ f(rm + sn) = r\, f(m) + s\, f(n)\]
\(-\) This is sometimes called an \(R-\) linear map.
\(-\) A bijective module homomorphism is called isomorphism of modules.
Types of Modules
- 1.
- Cyclic module: A module generate by a single elements.
- 2.
- Finitely generated: A module generated by a finite number of elements.
- 3.
- Free module: A module that has a basis or one which is isomorphic to some direct sum of some copies of \(R\).
- 4.
- Projective modules: Are direct summands of free modules.
- 5.
- Injective modules: Duals of projective modules.
- 6.
- Faithful module: \(M\) is faithful if the action of each \(r\neq 0\) in \(R\) on \(M\) in nontrivial (i.e, \(rx\neq 0\) for some \(x\in M\)).
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