Specified associative K-algebra A

A specified associative $K$ -algebra $A$ is an associative unital algebra with a fixed unital scalar map $K\to Z(A)$ , where $Z(A)$ is its center. Given a specified extension $K/F$ , composing scalar maps also makes $A$ an $F$ -algebra.

Four types of maps

A $K$ -module endomorphism is $K$ -linear; an $F$ -module endomorphism need only be $F$ -linear. A corresponding algebra endomorphism also preserves multiplication, with preservation of $1_A$ controlled separately by the category. The algebra need not be commutative.

Dimensions and comparison

When $[K:F]=d$ and $\dim_K A=m$ are finite, $\dim_F A=dm$ . Choosing bases identifies the two linear endomorphism rings with $M_m(K)$ and $M_{dm}(F)$ . Algebra endomorphisms are constrained by the multiplication table; dimension alone does not determine them. In infinite dimension the displayed dimension product is interpreted as cardinal multiplication.