Polynomial u-ring

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Not my strong suit but I've done the proofs for this

Definition

Let $R$ be a u-ring with the standard operations of $+$ and $*$. The set (or further-more: u-ring) of polynomials over $R$ (in the indeterminate $X$), denoted universally as $R[X]$, is defined as follows:

• Let $M:\eq\{e,X,X^2,X^3,\ldots,X^n,\ldots\}$ be the free monoid generated by $\{X\}$.
• A polynomial, $P\in R[X]$, over $R$ is a mapping: $P:M\rightarrow R$ by $P:X^n\rightarrow a_n$ such that $\vert\{a_n\in P(M)\ \vert\ a_n\neq 0\}\vert\in\mathbb{N}$
• The set of such polynomials is $R[X]$ - the set of all polynomials over $R$

Conventions

• It is normal to identify $r\in R$ with the constant polynomial: $a_n:\eq\left\{\begin{array}{lr}0 & \text{if }n>0\\r&\text{otherwise}\end{array}\right.$, which we can write more simply as $r$
  • Later we shall write this as $r+0X+0X^2+\cdots 0X^n+\cdots$ or more simply: $r$ (hence the "naturalness" of the identification)
• It is also normal to identify $X$ with the polynomial $a_n:\eq\left\{\begin{array}{lr}0 & \text{if }n\neq 1\\1&\text{otherwise}\end{array}\right.$
  • Later we shall write this as $0+1X+0X^2+\cdots+0X^n+\cdots$ or simply $X$ (hence the "naturalness" of the identification)

Notations

Claim 1: we may write $P\in R[X]$ as:

1. A summation: $\sum_{n\in\mathbb{N} }a_nX^n$ or more commonly: $\sum^\infty_{n\eq 0}a_nX^n$; or
2. $P:\eq a_0+a_1X+a_2X^2+\cdots+a_nX^n$ for some $n\in\mathbb{N}$.
   • Such an $n$ is actually called the degree of the polynomial Caveat:and can be defined only by convention if $P$ is the zero polynomial

U-ring operations

Claim 2: $R[X]$ is a u-ring itself. The operations are:

1. $P+Q$
2. $PQ$

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The terms are:

1. $(P+Q)_n\eq (P)_n+(Q)_n$ and
2. $(PQ)_n\eq \sum_{i+j\eq n}(P)_i(Q)_j$ - meaning the sum over terms whose indices add to $n$

Don't forget to flesh out the identity and unity!

Warning:That grade doesn't exist!

Identity (of the abelian additive group) and unity of the ring:

1. The zero polynomial: $0+0X+0X^2+\cdots+0X^n+\cdots$ which by the notation section we can simply write as: $0$
2. The unity of the ring is the polynomial $1$, (the thing we identify the unity of $R$ to).
   • More explicitly, the polynomial: $1+0X+0X^2+\cdots+0X^n+\cdots$

Proof of claims

Grade: D

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Done on paper:

• 

  Most of the work

See also

• Order (polynomial, polynomial uring)
• Degree (polynomial, polynomial uring)