K-algebra

𝕂-monoid

A 𝕂-monoid is an K-algebra for which the bilinear product possesses a two-sided identity 1 and is associative falg

• 1𝑎 =𝑎1 =𝑎 for all 𝑎 ∈𝐴
• (𝑎𝑏)𝑐 =𝑎(𝑏𝑐) for all 𝑎,𝑏,𝑐 ∈𝐴

Hence it is also called a unital associative algebra over 𝕂. In other words, a 𝕂-monoid is a field action on a ring. An algebra homomorphism which preserves the identity is a unital algebra homomorphism or 𝕂-monoid homomorphism. A generalization is an R-monoid.

Further terminology

• Algebraic element

Examples

• Matrix algebra over a field
• Complex number
• Quaternion (non-commutative)
• Endomorphism ring
• Tensor algebra
• Clifford algebra
• Extension field as a unital associative algebra

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