Polynomial over a ring

The usual definition of a polynomial is over a field as shown at polynomial over a field.

However, there is nothing in the immediate definition that prevents us from having a ring instead, i.e. a field but without the commutative property and inverse elements.

The only thing is that then we would need to differentiate between different orderings of the terms of multivariate polynomial, e.g. the following would all be potentially different terms:

2 x x y + 2 x y x + 2 y x x + x 2 x y + x 2 y x + y 2 x x + x x 2 y + x y 2 x + y x 2 x + x x y 2 + x y x 2 + y x x 2

(26)

while for a field they would all go into a single term:

12 x^{2} y

(27)

so when considering a polynomial over a ring we end up with a lot more more possible terms.

If the ring is a commutative ring however, polynomials do look like proper polynomials: Section "Polynomial over a commutative ring".

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Domain of a polynomial
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Numeric function
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Function
Formalization of mathematics
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Polynomial over a commutative ring