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 September 28th, 2015, 01:49 PM #1 Senior Member   Joined: Jan 2015 From: usa Posts: 104 Thanks: 1 localisation of ring Hi can someone please help me to prove this equivalence: Let A be a commutatif ring and a in A .we note Aa the localisation of A by S ={an, n dans N} knowing that A[x]/(ax-1) is isomorphic to Aa show that Aa #0 if and only if a is not nilpotent thaks in advance October 12th, 2015, 09:08 AM #2 Newbie   Joined: Oct 2015 From: Toronto Posts: 14 Thanks: 1 Suppose that $a$ is nilpotent, there exists an integer $n$ such that $a^n = 0$. Let $y$ be an element of $A[X]$, we denote by $\bar{y}$ the image of $y$ by the canonical surjection $A\rightarrow A_a =A[x]/(ax - 1)$. We have $\bar{a}\bar{x} =\bar{1}$. This implies that $\bar{1} = {\bar{a}}^n{\bar{x}}^n$. Since $a^n = 0$, we deduce that ${\bar{a}}^n =0$ and henceforth ${\bar{a}}^n{\bar{x}}^n =\bar{1} =0$. This is equivalent to saying that $A_a = 0$ since $\bar{1}$ is the identity of $A_a$. Suppose that $A_a=0$ this is equivalent to saying that $\bar{1} = 0$, or equivalently that $1\in (1-ax)$. We deduce the existence of a polynomial $P(x) = b_0 + b_1x+...+b_nx^n$ such that $(1-ax)P(x) = 1$. Lets determine the coefficients of $(1-ax)P(x)$ and deduce the coefficients of $P$ by using $(1-ax)P(x) = 1$. The coefficient of degree 0 of $(1-ax)P(x)$ is $1b_0 =1$ The degree 1 coeff. is $b_1-a = 0$, thus $b_1 = a$. Suppose that $b_i = a^i$, the $i+1$ coeff. of $(1-ax)P(x)$ is $b_{i+1} -a^{i+1} = 0$. We deduce that $b_{i+1} = a^{i+1}$ and recursively $b_n =a^n$. Since $(1-ax)P(x) = 1$, the biggest coefficient of $(1-ax)P(x)$ is 0.Since this coefficient is $a^{n+1}$, we deduce that $a^{n+1}=0$ and henceforth that $a$ is nilpotent. Last edited by Aristide Tsemo; October 12th, 2015 at 09:10 AM. Tags localisation, ring Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post limes5 Abstract Algebra 1 June 29th, 2015 12:24 AM Mathew Abstract Algebra 5 August 29th, 2010 08:53 PM tinynerdi Abstract Algebra 4 April 4th, 2010 10:17 PM John_maths Real Analysis 2 October 24th, 2008 02:05 PM cgouttebroze Abstract Algebra 5 August 14th, 2008 12:04 PM

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