Freese R., McKenzie R. Commutator Theory for Congruence Modular Varieties

Online version. — 2nd Ed. A revised. — (iv+174) p. English. Interactive menu.There is an operation naturally defined on the lattice of ideals of a ring, which has these properties. Now it develops, amazingly, that a commutator can be defined rather naturally in the congruence lattices of every congruence modular variety. This operation has the same useful properties that the commutator for groups (which is a special case of it) possesses.
The resulting theory has many general applications and, we feel, it is quite beautiful.The Commutator in Groups and Rings.
Exercise.
Universal Algebra.
Exercises.
Several Commutators.
Exercises.
One Commutator in Modular Varieties; Its Basic Properties.
Exercises.
The Fundamental Theorem on Abelian Algebras.
Exercises.
Permutability and a Characterization ofModular Varieties.
Exercises.
The Center and Nilpotent Algebras.
Exercises.
Congruence Identities.
Exercises.
Rings Associated With Modular Varieties: Abelian Varieties.
Exercises.
Structure and Representationin Modular Varieties.
Birkhoff-Jonsson Type Theorems For Modular Varieties.
Subdirectly Irreducible Algebras inFinitely Generated Varieties.
Residually Small Varieties.
Chief Factors and Simple Algebras.
Exercises.
Joins and Products of Modular Varieties.
Strictly Simple Algebras.
Mal’cev Conditions for Lattice Equations.
Exercises.
A Finite Basis Result.
Pure Lattice Congruence Identities.
The Arguesian Equation.
Related Literature.
Solutions To The Exercises:
Chapter (1-2, 4-10, 13).
Bibliography (83 publ.).