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 $R$