# What is difference between _free $\mathbb{F}$-algebra $A$_ and _vector space $V$ over $\mathbb{F}$_?

What is difference between free $$\mathbb{F}$$-algebra $$A$$ and vector space $$V$$ over $$\mathbb{F}$$?

Since the algebra is free, it has basis and so it is a vector space over $$\mathbb{F}$$.

So we have to look what more property the algebra $$A$$ hold.

Both the vector space $$V$$ and the algebra $$A$$ hold scalar multiplication and addition.

Also $$a,b \in A$$ implies $$ab \in A$$ but $$a, b \in V$$ does not imply $$ab \in V$$.

So this is the difference.

Am I right?

Now if I omit the freeness then what will be the difference between the algebra and the vector space ?

help me

$$a, b \in V$$ does not imply $$ab \in V$$.
It's more fundamental than this, though. It's not that $$ab$$ might happen to lie outside of $$V$$ if $$V$$ is a vector space. It's that $$ab$$ doesn't really make sense at all if $$V$$ is just a vector space.
So for instance, $$\Bbb C$$ is a free $$\Bbb R$$-vector space (with standard basis $$\{1, i\}$$), but not a free $$\Bbb R$$-algebra (using standard operations), because if we try a generating set like $$\{i\}$$, then $$1\in \Bbb C$$ may be written both as $$-i^2$$ and as $$i^4$$. And this kind of failure will happen no matter which generator we try.