User:Linus44

The plan
I'm a graduate student at Cambridge, UK, interested in algebra, algebraic geometry and number theory. I stumbled across the site and thought it was a neat idea. Since there aren't many results on commutative algebra I thought I'd start writing `commutative algebra corner' from the ground up. Basically I think a good graduate course worth of definitions and `little theorems' is needed. I hope that this will pave the way for writing a good resource for algebraic geometry and algebraic number theory.

Present Intentions
Organise existing results on undergraduate algebra on this page so I can link everything back nicely.

Definition:Polynomial Form
Let $$I_X=\left\{1,X,X^2,\ldots\right\}$$ be the  free monoid on a singleton $$\{X\}$$.

Let $$\left({R, +, \circ}\right)$$ be a ring with unity and additive identity $$0_R$$.

A polynomial form in one variable, or just polynomial over $$R$$ is a mapping $$f:I_X\to R:X^n\mapsto a_n$$ such that $$a_n=0$$ for all but finitely many $$n\geq 0_R$$.

Therefore a polynomial is an ordered triple $$(R,I_X,f)$$. We describe the polynomial as in the indeterminate $$X$$. Often, this singleton is unimportant, and we speak simply of the polynomial $$f$$ over the ring $$R$$.

Definition: Indeterminate
Redirect to Definition: Polynomial Form

Definition:Degree (Polynomial Form)
Let $$f$$ be a polynomial form over a ring $$R$$ in the indeterminate $$X$$. The degree of $$f$$ is


 * $\deg(f)=\sup\left\{n\in\N:\Delta(X^n)\neq 0\right\}$

Representation of Polynomials
Theorem

Let  $$I_X=\left\{1,X,X^2,\ldots\right\}$$ be the   free monoid on a  singleton $$\{X\}$$.

Let $$\left({R, +, \circ}\right)$$ be a ring with unity.

Let $$P$$ be the set of all polynomials over $$R$$ in the indeterminate $$X$$. That is, the set of all ordered triples $$(R,I_X,f)$$ where $$f:I_X\to R$$ is such that $$f(X^i)=0$$ for all but finitely many $$i\geq 0$$

Let $$rX^n$$ be the map that equals $$r$$ on $$X^n$$ and $$0$$ otherwise.

Let $$S$$ be the set of all formal sums of the kind


 * $r_0+r_1X+\cdots +r_nX^n$

where $$r_i\in R$$.

Let $$F:P\to S$$ be the map under which the polynomial $$f:X^i\mapsto a_i$$ is sent to the formal sum $$a_0+a_1X+\cdots+a_dX^d$$, where $$d=\deg(f)$$.

Then $$F$$ is an isomorphism of rings.

Representation of Polynomials
Theorem

Let $$f$$ be a polynomial form over a ring $$R$$ in the indeterminate $$X$$.

Let $$d=\deg(f)$$.

For $$r\in R$$, let $$rX^n$$ denote the function that equals  $$r$$ on $$X^n$$ and $$0$$ otherwise. Then each polynomial can be written uniquely as a sum


 * $r_0+r_1X+\cdots +r_dX^d$

for some $$r_n\in R$$. Furthermore if $$d'd$$. Then for each $$n\leq d$$, we have


 * $f(X^n)=a_n$,

and


 * $(a_0+a_1X+\cdots +a_dX^d)(X^n)=0+\cdots+a_n+\cdots+0=a_n$. Additionally, both $$f$$ and $$a_0+a_1X+\cdots +a_dX^d$$ are zero on $$X^n$$, $$n>d$$.

Therefore $$f=a_0+a_1X+\cdots +a_nX^n$$.

Now suppose that also $$f=a_0'+a_1'X+\cdots+a_{d'}'X^{d'}$$. Then $$f(X^d)=a_d\neq 0$$, but $$(a_0'+a_1'X+\cdots+a_{d'}'X^{d'})(X^d)=0$$, a contradiction.

Definition: Polynomial Addition
Let $$(R,I_X,f)$$ and $$(R,I_X,g)$$ be polynomials in $$X$$ over the ring $$R$$. Addition of polynomials is defined by



Ring theory
Rings

Definition:Ring

Definition:Unity

Definition:Ring with Unity

Definition:Commutative Ring

Definition:Commutative Ring With Unity

Characteristic of a Ring

Binomial Theorem

Subrings and Ideals

Definition:Subring

Definition:Ideal

Definition:Generating Set

Definition:Principal Ideal

Definition:Maximal Ideal

Definition:Sum of Ideals

Definition:Quotient Ring

Intersection of Ideals

Increasing Union of Ideals is Ideal

Sum of Ideals is an Ideal

Commutative Quotient Ring

Quotient Ring with Unity

Natural Epimorphism to Quotient Ring

Homomorphisms

Definition:Homomorphism

Definition:Monomorphism

Definition:Epimorphism

Definition:Isomorphism

Definition:Kernel

Definition:Image

Ring Homomorphism Preserves Subrings

Ring Epimorphism Inverse of Subring

Ring Epimorphism Preserves Ideals

Ring Epimorphism Inverse of Ideal

Kernel of Ring Homomorphism is Ideal

First Isomorphism Theorem

Domains and fields

Definition:Integral Domain

Definition:Field

Definition:Prime Ideal

Definition:Maximal Ideal

Definition:Quotient Field

Characteristic of Ring with No Zero Divisors

Prime Ideal iff Quotient Ring is Integral Domain

Maximal Ideal iff Quotient Ring is Field

Maximal Ideal is Prime

Field Homomorphism is Injective

Existence of Quotient Field

Quotient Field is Unique

PIDs, UFDs, Noetherian rings

Galois theory
Polynomials in one variable

Definition:Polynomial

Definition:Ring of Polynomial Forms

Definition:Degree (Polynomial)

Set of Polynomials is a Subring

Ring of Polynomial Forms is Integral Domain

Polynomials Closed under Ring Product

Rings of Polynomial Forms Isomorphic

Unique Representation in Polynomial Forms

1-var polynomial rings, Fields, extensions, algebraic extensions, algebraic clusure, separable extensions, splitting fields, normal extensions, galois extensions, polynomials, cyclotomy, norm and trace.

Commutative algebra
Definitions

Definition:Jacobson Radical Definition:Nilpotent Element Definition:Nilradical

Definition:Radical Ideal

Definition:Radical of an Ideal

Definition:Reduced Ring

Little theorems

Characterisation of the Jacobson Radical

Cayley-Hamilton Theorem

Hilbert's Basis Theorem

Nakayama's Lemma

Radical Ideal iff Quotient Ring is Reduced