Definition. Let K be a field. An algebra A over K is a set on which three operations are defined, addition, multiplication and multiplication by a scalar such that
i) A is a ring under addition and multiplication,
ii) A is a vector space over K under addition and multiplication by a scalar,
iii) For all $a \in K$, $x,y, \in A$, $(ax)y=x(ay)=a(xy)$.
As an exercise. If A is a nonzero K-algebra, check that $\{a 1_{a}: a \in K\}$ is a subring of A isomorphic to K. Since have $ax=(a 1_{a})x$ for $x \in A$, $a \in K$
See I'm trying to understand what it mean's a K-algebra. Does it just mean an algebra over K?
Also, what does i) mean i.e. what is an vector space over K? Does it mean a vector space if you take the scalars as the non-zero elements of K?