Field/Algebra/Directly/Definition

Algebra

Let $K$ be a field, and let $A$ denote a $K$-vector space. $A$ is called a commutative $K$-Algebra if there exists a fixed element $1$ and a binary operation, called multiplication,

$A\times A⟶A,(a,b)⟼a\cdot b,$

such that the following conditions hold.

1. We have

   $1\cdot a=a$

   for all $a\in A$.

2. The operation is associative.
3. We have

   $a\cdot b=b\cdot a$

   for all $a\in A$.

4. For $c\in K$ and $a\in A$, we have

   $ca=(c1)\cdot a,$

   where $ca$ and $c1$ means the scalar multiplication.