Algebra theory MOC

Algebra over a field

An algebra over a field is a Vector space over equipped with a bilinear product , falg i.e. for any and

This may be generalized to an R-algebra.

Examples

• K-monoid^[In these notes I will try and reserve infix notation for associative algebras, as there is a tendency to assume such things to be associative.]
  • Commutative
    • Complex number
    • Quaternion (non-commutative)
    • Symmetric algebra
  • Non-commutative
    • Matrix algebra over a field
    • Endomorphism ring
    • Tensor algebra
    • Clifford algebra
    • Monoid ring
    • Extension field as a unital associative algebra
• Category ring
• Non-associative
  • Lie algebra

Properties

• The product within an algebra is completely determined by its Structure constants

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