k-algebra

Noun

k-algebra (plural k-algebras)

1. (algebra) An algebra over a field; a ring with identity together with an injective ring homomorphism from a field, k, to the ring such that the image of the field is a subset of the center of the ring and such that the image of the field’s unity is the ring’s unity.

   A k-algebra A with ring homomorphism ${\displaystyle \phi :k\rightarrow A}$ is a k-vector space with scalar multiplication (i.e., action of k upon A): ${\displaystyle \lambda \cdot a=\phi (\lambda )a}$ where ${\displaystyle \lambda \in k,\ a\in A}$.