Element with divided powers

Definition
Suppose $$R$$ is an associative ring. An element with divided powers in $$R$$ is a sequence:

$$\mathbf{x} = (x^{(1)},x^{(2)},x^{(3)},\dots,x^{(n)},\dots)$$

with the property that:

$$\! x^{(i)}x^{(j)} = \binom{i + j}{i} x^{(i+j)}$$

By abuse of notation, we sometimes simply say that the first element of the sequence, $$x^{(1)}$$, is the element with divided powers (see the notes below).

Alternatively, we can define an element with divided powers as a homomorphism from (the ideal generated by positive degree elements in the free divided power algebra in one variable) to $$R$$.

In the case that $$R$$ is an associative unital ring, we can define $$x^{(0)} = 1$$. In this case, we can define an element with divided powers as a homomorphism from the free divided power algebra in one variable to $$R$$.

Facts

 * If the additive group of $$R$$ is torsion-free, then any element with divided powers is determined uniquely by its first coordinate. Explicitly, if $$\mathbf{x} = (x^{(1)},x^{(2)},\dots)$$ and $$\mathbf{y} = (y^{(1)},y^{(2)},\dots)$$ are elements with divided powers and $$x^{(1)} = y^{(1)}$$, then $$\mathbf{x} = \mathbf{y}$$. Note, however, that not every element of $$R$$ can be the first coordinate of an element with divided powers, because its powers may not admit division.
 * If $$R$$ is an algebra over the field of rational numbers, then every element of $$R$$ occurs as the first coordinate of a unique element with divided powers in $$R$$. Explicitly, there is a bijection:

The underlying set of $$R$$ $$\leftrightarrow$$ The elements with divided powers in $$R$$