Examples of algebras having a module basis

Question

I'm looking for examples of associative $R$-algebras, for which an $R$-module basis can be specified.

Of course, if $K$ is a field, then any $K$-algebra admits such a basis, but this dis not what I'm looking for. Such a basis may be highly elusive, e.g. the basis of the formal power series algebra $K[[x_1,\ldots,x_n]]$ is nowhere to be seen.

Clearly semigroup algebras $R[S]$ and path algebras of a multidigraph $R[\Gamma]$ are such examples. If algebra $A$ has a module basis, then so too does the matrix algebra $M_n(A)$. Also, a polynomial algebra with monomial relations $R[x_1,\ldots,x_n]/(x^{a_1},\ldots,x^{a_k})$ has a basis $\{x^a; a\!\in\!\mathbb{N}^n, a_i\!\nleq\!a\text{ for all }i\}$, where $\leq$ is the componentise partial order. Any other examples?

Answer 1

1. Let $\underline{X}$ be a small category. Then its morphism algebra $R[\underline{X}]$ is the $R$-module with basis the morphisms of $\underline{X}$ and multiplication the composition of morphisms (if end of $\alpha$ is not the beginning of $\beta$, then $\alpha\beta=0$). This is an associative algebra, which is unital iff $\underline{X}$ has finitely many objects (then the unity $1$ of $R[\underline{X}]$ is the sum of all identity morphisms $1_A$ of $\underline{X}$).

2. Semigroup algebras, path algebras, incidence algebras are special cases of $R[\underline{X}]$ (when $\underline{X}$ has only one object, when $\underline{X}$ is a free category on a digraph, when $\underline{X}$ has at most one arrow between any two objects).

3. If $f$ is any monic polynomial of degree $n$, then $R[x]/(f)$ has basis $\{1,x,\ldots,x^{n-1}\}$.