Talk:Commutative algebra

On binary commutating structures

Binary numbers forming a commutating field are the complex numbers, but there are also other commutating binary (or two component) numbers, which form rings, rather than fields. (The inverse does not exist for all nonzero elements)...but does exist for most.

For complex numbers

Z=x+ty where tt = -1

For ring of dual numbers

Z=x+ty where tt = 0

For perplex numbers

Z=x+ty where tt = 1

A nifty property of the ring of dual numbers is to express a function over this domain as f(z) = f(x+ty) = f(x)+ty df/dx where tt=0. This follows from Taylor. Gives a nice algebraic definition of derivative.

Generalized Cauchy Riemann relations also exists for functions over the new domains.

Questions: Is this delving into homological algebra? what can the interplay of algebra and analisis be called? How is this classified?

astarfish — Preceding unsigned comment added by 200.9.237.254 (talk) 14:08, 11 July 2005 (UTC)

Adding Macaulay2 Computations

I think the algebraic geometry, commutative algebra, and algebraic number theory pages on wikipedia should include example computations using computer algebra systems. The main programs I know of are macaulay2, sage, and singular. Does anyone have thoughts about this? — Preceding unsigned comment added by 128.138.65.151 (talk) 21:41, 28 August 2017 (UTC)

Todo

• Cite the paper mentioned in https://mathoverflow.net/questions/21067/noetherian-rings-of-infinite-krull-dimension
• This page should include the Nullstellensatz after the Hilbert Basis theorem and it should be mentioned how this is foundational to commutative algebra.
• The nullstellensatz should be used to motivate the functor of points and the definition of an affine scheme as a co-representable functor on ${\textbf {CRing}}$.
• The connections with algebraic geometry section needs to be expanded further. This should include mentioning graded rings with (weighted) projective schemes/DM-stacks. — Preceding unsigned comment added by 70.59.20.131 (talk) 02:32, 29 August 2017 (UTC)