Noun
k-algebra (plural k-algebras)
(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
ϕ
:
k
→
A
{\displaystyle \phi :k\rightarrow A}
is a k-vector space with scalar multiplication (i.e., action of k upon A):
λ
⋅
a
=
ϕ
(
λ
)
a
{\displaystyle \lambda \cdot a=\phi (\lambda )a}
where
λ
∈
k
,
a
∈
A
{\displaystyle \lambda \in k,\ a\in A}
.
Since the center of a simple k-algebra is a field, any simple k-algebra is a central simple algebra over its center. Source: Internet
Also, we mostly fix the base field; thus, an algebra refers to a k-algebra. Source: Internet