Properties
It can be verified that:
- Formal differentiation is linear: for any two polynomials f(x), g(x) and elements r, s of R, we have
- When R is not commutative there is another, different linearity property in which r and s appear on the right rather than on the left. When R does not contain an identity element then neither of these reduces to the case of simply a sum of polynomials or the sum of a polynomial with a multiple of another polynomial, which must also be included as a "linearity" property.
- The formal derivative satisfies the Leibniz rule, or product rule:
- Note the order of the factors; when R is not commutative this is important.
These two properties make D a derivation on A (see also module of relative differential forms for a discussion of a generalization).
Read more about this topic: Formal Derivative
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