View source for Polynomial u-ring

{{Stub page|grade=A|msg=Not my strong suit but I've done the proofs for this}}
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==Definition==
Let {{M|R}} be a [[u-ring]] with the standard operations of {{M|+}} and {{M|*}}. The ''set (or further-more: u-ring) of polynomials over {{M|R}}'' (in the ''indeterminate'' {{M|X}}), denoted universally as {{M|R[X]}}, is defined as follows{{rAAPAG}}:
* Let {{M|M:\eq\{e,X,X^2,X^3,\ldots,X^n,\ldots\} }} be the [[free monoid generated by]] {{M|\{X\} }}.
* A ''polynomial'', {{M|P\in R[X]}}, over {{M|R}} is a [[mapping]]: {{M|P:M\rightarrow R}} by {{M|P:X^n\rightarrow a_n}} such that {{M|\vert\{a_n\in P(M)\ \vert\ a_n\neq 0\}\vert\in\mathbb{N} }}<ref group="Note">This may be said as:
# {{M|a_n\eq 0}} for ''[[almost all]]'' {{M|n\in\mathbb{N}_0}} (as per one of the references - Abstract Algebra: Grillet),
# {{M|a_n\eq 0}} [[almost always]], or
# {{M|\vert\{a_n\in P(M)\ \vert\ a_n\eq 0\}\vert\eq\aleph_0}}
#* But this requires a few things: [[the cardinality of the free monoid generated by a single object is countable]] and [[finitely many elements removed from a countable set remains countable]]
I have written: {{M|\vert\{a_n\in P(M)\ \vert\ a_n\neq 0\}\vert\in\mathbb{N} }} - which is quite literally "there are finitely many elements that are the additive identity of {{M|R}}"</ref>
* The set of such polynomials is {{M|R[X]}} - ''the set of all polynomials over {{M|R}}''
===Conventions===
* It is normal to identify {{M|r\in R}} with the ''constant polynomial''<ref name="AAPAG"/>: {{M|a_n:\eq\left\{\begin{array}{lr}0 & \text{if }n>0\\r&\text{otherwise}\end{array}\right.}}, which we can write more simply as {{M|r}}
** Later we shall write this as {{M|r+0X+0X^2+\cdots 0X^n+\cdots}} or more simply: {{M|r}} (hence the "naturalness" of the identification)
* It is also normal to identify {{M|X}} with the polynomial {{M|a_n:\eq\left\{\begin{array}{lr}0 & \text{if }n\neq 1\\1&\text{otherwise}\end{array}\right.}}
** Later we shall write this as {{M|0+1X+0X^2+\cdots+0X^n+\cdots}} or simply {{M|X}} (hence the "naturalness" of the identification)
===Notations===
'''Claim 1: ''' we may write {{M|P\in R[X]}} as:
# A summation: {{M|\sum_{n\in\mathbb{N} }a_nX^n}} or more commonly: {{M|\sum^\infty_{n\eq 0}a_nX^n}}; or
# {{M|P:\eq a_0+a_1X+a_2X^2+\cdots+a_nX^n}} for some {{M|n\in\mathbb{N} }}.
#* Such an {{M|n}} is actually called the [[degree (polynomial, polynomial uring)|degree of the polynomial]] {{caveat|and can be defined only by convention if {{M|P}} is the zero polynomial}}
===U-ring operations===
'''Claim 2: ''' {{M|R[X]}} is a [[u-ring]] itself. The operations are:
# {{M|P+Q}}
# {{M|PQ}}
{{Requires work|grade=B|msg=The terms are:
# {{m|(P+Q)_n\eq (P)_n+(Q)_n}} and
# {{m|(PQ)_n\eq \sum_{i+j\eq n}(P)_i(Q)_j}} - meaning the sum over terms whose indices add to {{M|n}}
Don't forget to flesh out the identity and unity!}}
Identity (of the [[abelian group|abelian]] additive [[group]]) and unity of the ring:
# The zero polynomial: {{M|0+0X+0X^2+\cdots+0X^n+\cdots}} which by the notation section we can simply write as: {{M|0}}
# The unity of the ring is the polynomial {{M|1}}, (the thing we identify the unity of {{M|R}} to).
#* More explicitly, the polynomial: {{M|1+0X+0X^2+\cdots+0X^n+\cdots}}
==Proof of claims==
{{Requires proof|grade=D|msg=Done on paper:
<gallery>
Image:Polynomial ring work.JPG|Most of the work

</gallery>}}

==See also==
* [[Order (polynomial, polynomial uring)]]
* [[Degree (polynomial, polynomial uring)]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Abstract Algebra|Ring Theory}}