For a unital ring R, the free-R-module monad on sets has left R-modules as its Eilenberg–Moore algebras

By [L1]–[L2], the monad sends $X$ to finite formal sums ${\sum }_x{r}_x[x]$, its unit sends $x$ to $1[x]$, and multiplication evaluates a finite formal sum of finite formal sums. Every left $R$-module gives an algebra by evaluating such sums.

Conversely, for an algebra $a:R^{(X)}\to X$, define $0=a(0)$, $x+y=a(1[x]+1[y])$, and $rx=a(r[x])$. The algebra unit and multiplication laws say that substituting finite formal sums before evaluation has the same result as flattening them first; the module axioms in [L3] follow by applying this equality to the corresponding formal sums.

An algebra homomorphism commutes with evaluation, hence preserves $0$, addition, and scalar multiplication. Conversely an $R$-linear map preserves every finite formal sum and is therefore an algebra homomorphism. These identifications are inverse over $\mathbf{S}\mathbf{e}\mathbf{t}$.