Definition:Algebra over Field/Also defined as

Algebra over Field: Also defined as

Some sources insist that an algebra over a field requires that the bilinear mapping $*$ must have an identity element $1_A$ such that:

$\forall a \in A: a * 1_A = 1_A * a = a$

that is, that $\struct {A, *}$ has to be a unitary algebra.

It is worth being certain of what is meant in any works read.

Especially in commutative algebra, an algebra over a field is often defined as a unital associative commutative algebra.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): algebra over field
• John C. Baez: The Octonions (2002): 1.1 Preliminaries