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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?

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Let $A$ be a ring in the usual sense. It has a multiplication and an addition that satisfy all the distributive and associative laws and such. But, now suppose that $A$ is also a vector space over the field $K$. So for every $k\in K$ and $a\in A$ you have a new element $ka$. Now, the issue arises that for another element $b\in A$ you have that $kb\in A$. Your $(iii)$ says that $(ka)b=a(kb)$. In other words $k$ is in the center of $A$; it commutes with all elements of $A$. To answer you last question it is a vector space if you take all the elements of $K$ as the scalars, including $0$.

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    @simplicity: By definition of vector space $ka\in A$.2011-11-29