Definition of associative algebra over a field

Question

In the definition of an algebra over a field in the wiki entry , it states that an algebra over a field is a vector space equipped with a bilinear product.

Question: Does anyone know how a bilinear product is defined?

Then a unital associative algebra is defined: An algebra such that the product is associative and has an identity is therefore a ring that is also a vector space, and this equipped with a field of scalars. Such an algebra is called here a unital associative algebra.

Question: It is stated that the associativity of the product and identity makes it a ring that is also a vector space. What about the left and right distributivity of multiplication over addition which is required for a ring? How is that guaranteed to be satisfied?

Answer 1

Let $A$ be a vector space over $\Bbb{K}$.

By a bilinear product we mean a map (the "multiplication") $m: A \times A \to A$ so that for all $x,y,z \in A$ and $\alpha, \beta \in \Bbb{K}$, $$m(\alpha x + \beta y, z) = \alpha m(x, z) + \beta m(y, z)$$ and similarly $$m(x, \alpha y + \beta z) = \alpha m(x, y) + \beta (x, z),$$ in other words, it is linear in both "components".

But this "bilinearity" is precisely what allows you to multiply over sums. If in addition this bilinear product is associative and has an identity, this means precisely that $A$ is a now a ring.

Answer 2

Bilinear product means that the map "product" is bilinear, so $(a,b) \mapsto ab$ is a bilinear map.

For the second question: these are consequences of the product being bilinear.