Core Structure Theory
Modules and R-Modules as Algebras
Modules over a fixed ring as algebras with one unary operation per scalar, and the sense in which modules are the model case for the whole structure theory of congruence-modular varieties.
Learning objectives
- Present R-modules in the universal-algebraic format
- Show that submodules correspond to congruences
- Explain why modules are the reference point for commutator theory
The type
For a fixed ring R with unit, a left R-module is an algebra ⟨M, +, −, 0, (fr)r∈R⟩ where ⟨M, +, −, 0⟩ is an abelian group and each fr is a unary operation representing multiplication by r.
The module axioms become identities in this type:
- fr(x + y) ≈ fr(x) + fr(y)
- fr+s(x) ≈ fr(x) + fs(x)
- frs(x) ≈ fr(fs(x))
- f1(x) ≈ x
If R is infinite the type has infinitely many operation symbols. Nothing in the general theory requires a finite type; only finitary arity is required, and every fr is unary.
Congruences are submodules
For an R-module M, the map sending a submodule N to the congruence {⟨a, b⟩ : a − b ∈ N} is a lattice isomorphism from the submodule lattice onto Con(M).
This is the module version of a phenomenon shared by groups (normal subgroups) and rings (two-sided ideals): congruences are determined by a single class, the one containing the identity element.
Because these varieties have a Mal'cev term: p(x, y, z) = x − y + z satisfies p(x, y, y) ≈ x and p(x, x, z) ≈ z. Mal'cev's theorem says this is exactly the condition for a variety to be congruence-permutable, and permutability is what forces congruences to be determined by one class.
Modules as the model case
Modules are congruence-modular, congruence-permutable, and have a well-understood commutator. This combination makes them the reference against which general structure theory is measured.
Chapter II §13 develops the centre of an algebra and proves a characterisation of modules up to polynomial equivalence — a result showing that the module case can be recognised intrinsically, without reference to a ring.
Abelian algebras
An algebra A is abelian if for every term t and all tuples, t(a, c) = t(a, d) implies t(b, c) = t(b, d) for all a, b.
Modules are abelian in this sense, and the condition captures precisely what is module-like about them. In a congruence-modular variety, an abelian algebra is polynomially equivalent to a module over a ring — a result due to work after the source text, and one of the most striking in the subject.
The general theorem identifying abelian algebras in congruence-modular varieties with modules belongs to the commutator theory developed largely after 1981, principally by Freese, McKenzie, Gumm and Hagemann–Herrmann. The source text develops the centre in §13 and points forward to this line of work in its closing chapter.
Frequently asked questions
Why not treat scalar multiplication as a single binary operation?
Because one of its arguments comes from R, not from M, so it is not an operation on the module's universe. Splitting it into a family of unary operations keeps everything inside the type, at the cost of a possibly infinite signature.
Are vector spaces a variety?
Yes, for a fixed field. A vector space over a fixed field K is a K-module in this sense, and the class is defined by identities. What is not a variety is the class of all vector spaces over all fields.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section II.1, book pages 28-30.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.