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I have struggled to define "Polynomial Ring" today. Since I'm not familiar with abstract algebra, i don't know if there is a theorem states that "For every commutative ring $R$ with unity, there exists a topology on $R$".

I'm wondering this, because i think, to directly define $R[x]$, $\sum_{k=0}^\infty a_k x^k$ should be defined first for arbitrary sequence $a$, that is, limit should be defined first.

To avoid this, i first defined binary operations on a set $I$ of sequences $\sigma$ in $R$ such that $\sigma^{-1}(R\setminus \{0\})$ is finite. (In the usual way) Then I showed $I$ is a commutative ring with unity.

Then define a homomorphism to define $R[x]$. (That is, define $f(a)=\sum_{k=0}^n a_k x^k$ such that $n=\max\{i\in\omega:a_i \neq 0\}$ for all $a\in I$ and let $R[x]\triangleq f(I)$).

Is my approach O.K? Or if there is a nice way to define $R[x]$, please let me know..

Katlus
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  • You might be interested in the definition here: http://planetmath.org/PolynomialRing.html Using limits/convergence to construct polynomial rings is overkill. – Adam Saltz Feb 05 '13 at 05:38

2 Answers2

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The ring $R[x]$ can be formally defined as follows.

The elements of $R[x]$ are all infinite sequences $(r_0,r_1,r_2,\dots)$ such that all but a finite number of the $r_i$ are $0$. The sum of two such sequences is defined in the natural way.

The product is the convolution product. Let $(a_0,a_1,a_2,\dots)$ and $(b_0,b_1,b_2,\dots)$ be two such sequences. Their product is defined as $(c_0,c_1,c_2,\dots)$, where $$c_n=\sum_{i=0}^n a_i b_{n-i}.$$ It turns out that the object $(0,1,0,0,0,\dots)$ behaves like $x$ should. The polynomial ordinarily called $r_0+r_1x+r_2x^2+\cdots+ r_nx^n$ can be identified with the sequence $(r_0,r_1,\dots, r_n,0,0,0,\dots)$.

André Nicolas
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  • Aha! I wonder why author of my textbook didn't write that one simple word 'sequence'.. Thank you – Katlus Feb 05 '13 at 05:50
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    Note also that if you allow unrestricted sequences, you obtain the ring of formal power series $R[[x]]$. Elements of $R[[x]]$ are sequences $(a_0,a_1,\ldots)$ in $R$ that we denote as $\sum_{k=0}^\infty a_k x^k$. However, the summation notation should NOT be mistaken for the addition operation in $R[[x]]$. – John Myers Feb 05 '13 at 05:52
  • @John: ... although it is a convergent series in the usual topology on $R[[x]]$. –  Feb 05 '13 at 08:02
  • @Hurkyl: The OP mentioned trying to make sense of $\sum^\infty_{k=0} a_k x^k$ in an arbitrary commutative ring. My only point is that we don't need a topology beforehand to define $R[[x]]$ (via sequences in $R$). – John Myers Feb 05 '13 at 08:24
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One simple and natural definition of $\rm\,R[x]\,$ is the subring of linear maps on $\rm\,R^\Bbb N\,$ generated by the shift map $\rm\,x\!:\: (r_0,r_1,\ldots)\to (0,r_0,r_1,\ldots)\:$ and scalings $\rm\:r\!:\: (r_0,r_1,\ldots)\to (rr_0,rr_1,\ldots),\ r\in R.$

Remark $\ $ Note that $\rm\,x\,$ is transcendental over the subring of scaling maps $\rm(\cong R),$ since

$$\rm\ (r_0 + r_1\, x + \cdots + r_n\, x^n)\, (1,0,0,\ldots)\ =\, (r_0,r_1,\ldots, r_n,0,0,\ldots)$$

which is nonzero when the polynomial is nonzero.

Math Gems
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