📂Abstract Algebra

Partial Differential Rings and Differential Rings

Definition ¹

In a ring $R$, if the algebraic derivatives $\Delta = \left\{ \partial_{1} , \cdots , \partial_{n} \right\}$ defined satisfy $$ \partial_{i} \left( \partial_{j} (r) \right) = \partial_{j} \left( \partial_{i} (r) \right) \qquad \forall r \in R $$ for all $i,j = 1, \cdots, n$, then the ordered pair $\left( R , \Delta \right)$ is called a Partial Derivative Ring. Especially, if $\Delta$ is a Singleton set, that is $\Delta = \left\{ d \right\}$, then $\left( R , \Delta \right) = \left( R , d \right)$ is called an Ordinary Derivative Ring.

Description

Having seen that differential equations are broadly divided into ordinary differential equations and partial differential equations, it is natural that people studying algebraic derivatives initially distinguished between ordinary and partial derivatives. What particularly distinguishes it from differentiation in analysis is the condition that they commute as in $\partial_{i} \partial_{j} = \partial_{j} \partial_{i}$. While this usually relates to properties of continuity, it is peculiar how it is starkly defined in algebra.