Special linear group over a commutative unital ring
Let be a commutative unital ring and be a natural number. The special linear group of degree over , denoted or , is defined as the subgroup of the general linear group comprising those matrices whose determinant is . Equivalently, it is the kernel of the determinant homomorphism.
Here, the determinant for a matrix is defined in the usual way as a polynomial function. Note that this function is independent of the choice of basis, hence the special linear group can be considered more abstractly for any free module over without an explicit basis.
- Special linear group over a field is the typical use.
- Special linear group over a unital ring: A generalization based on a corresponding generalization of the notion of determinant to the non-commutative case.
- Special linear group over a division ring
- Group generated by elementary matrices over a unital ring. This is contained in the special linear group, but need not be the whole special linear group. When the ring is a Euclidean domain or a division ring, it coincides with the whole group.
- Steinberg group over a unital ring: This has a natural homomorphism surjecting to the group generated by elementary matrices. The homomorphism is an isomorphism in the case of fields.