Modules: Linear Algebra Over a Ring
A module is a vector space whose scalars form a ring rather than a field. Bases may fail to exist, and what survives is a structure theorem that delivers the Jordan form and the classification of finite abelian groups at the same time.
What a learner can do afterwards
- Give a module with no basis and say which vector-space step fails
- State the structure theorem for finitely generated modules over a principal ideal domain
- Read the Jordan form of a matrix as one instance of that theorem
1 · Read
You already know vector spaces, where scalars come from a field and every nonzero scalar has an inverse. A module keeps the same addition rules but lets scalars come from a ring, so division can fail. That failure creates torsion: a nonzero scalar times a nonzero element can be zero. Integers mod 2 as a module over integers show this, since 2 times anything is 0, so no independent set exists and the module has no basis.
You compare this with a free case. Pairs of integers form a free module of rank 2 with basis (1, 0) and (0, 1). Vector spaces always have bases, but modules need not, and the step that breaks is invertibility of nonzero scalars. Integers mod 6 show the same failure, since 2 times 3 is 0 mod 6 with both factors nonzero, so independence fails there too.
You then use the structure theorem for finitely generated modules over a principal ideal domain. It splits each module into a free part plus torsion cyclic pieces of prime power order. So integers mod 6 split as cyclic of order 2 plus cyclic of order 3, and sizes multiply to give 4 for two order 2 pieces and 24 for orders 4 and 6. Over polynomials the same theorem gives Jordan form: eigenvalue 5 with one eigenvector means one block with 5 on the diagonal and 1 above.
You allow ring scalars, watch torsion block bases, and split modules into free rank plus cyclic torsion, which also gives Jordan form.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.