Polynomial ring indexed by an arbitrary set.

Question

Let $B$ be any non-empty set, possibly uncountable. What does the term a polynomial ring indexed by the set $B$ means?

Answer 1

Just like $k[X]$ is "the most general commutative $k$-algebra generated by an element $X$", $k[X_b]_{b\in B}$ is the most general commutative $k$-algebra generated by elements $X_b$ indexed by $B$.

It elements are (finite) formal combinations of monomials $X_{b_1}^{\alpha_1}\cdots X_{b_r}^{\alpha_r}$ where the $X_b$ are your indeterminates (with $b\in B$). The multiplication is the obvious one : you multiply the monomials by adding the powers, and you extend that to general polynomials by bilinearity.

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Formally, if $\mathbb{N}^{(B)}$ is the set of functions $B\to \mathbb{N}$ with finite support (ie only a finite number of elements of $B$ have non-zero image), then $k[X_b]_{b\in B}$ may be constructed as the set of maps $\mathbb{N}^{(B)}\to k$ with finite support, with its usual $k$-vector space structure, and the product $(f\ast g)(x) = \sum_{a+b=x} f(a)g(b)$.

It can also be characterized as the free $k$-algebra on the set $B$, which means it satisfies the universal property that for any commutative $k$-algebra $A$, and any function $f:B\to A$, there is a unique extension of $f$ to a $k$-algebra morphism $\tilde{f}:k[X_b]_{b\in B}\to A$ (satisfying $\tilde{f}(X_b) = f(b)$).