Algebra Over A Field - Algebras and Rings

Algebras and Rings

The definition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field K is a ring A together with a ring homomorphism

where Z(A) is the center of A. Since η is a ring morphism, then one must have either that A is the trivial ring, or that η is injective. This definition is equivalent to that above, with scalar multiplication

given by

Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as

for all and . In other words, the following diagram commutes: