What would an element of an algebra generated by a subalgebra look like?

Question

If we have $A$ is an algebra and $X \subset A$ generates $A$, what does that look like? Also, if we have an algebra generated by a single element what would an element in that algebra look like?

Answer 1

For an $R$-algebra $A$, we say that $X \subset A$ is a generating subset if for any element $a \in A$ we can find an element $r_a$ in $R[[X]]$ (formal power series with coefficients in $R$ and variables in $X$) such that $a=r_a$ (consider the power series as already evaluated in $X$, applying the algebra operations).

In the special case when $X=\{x_1, \dots, x_n \}$ is finite, we say that $A$ is finitely generated if $r_a \in R[x_1, \dots, x_n]$, that is, if $r_a$ is a polynomial.

An example of an algebra generated by a single element is $\mathbb{K}[X]$, the ring of polynomials over a field.