Let $R$ be a ring and let $x , y$ be elements of $R$ with $xy=1,yx\neq1$.
Let $S$ be the set of all elements $w\in R$ such that $xw=1$.
How I can proof that $S$ is an infinite set ,$($considering $w_n=y+(yx-1)x^n$ ,for $n=1,2,3,\cdots )$$??$
Asked
Active
Viewed 266 times
2
Jhwana
- 535
-
Is the general case correct? ((Any ring $R$ containing some elements $x,y$ such that $xy=1$ and $yx\neq1$ must be infinite,In other word: If $R$ is finite and $xy=1$ then $yx=1$ )) – Jhwana Mar 17 '13 at 21:42
2 Answers
2
OK, so the above suggests we consider $xw_n$, for any given $n$, right?
Computing...
$$ xw_n = xy + xyx^{n+1} - x^{n+1} = 1 + x^{n+1} - x^{n+1} = 1 $$
which holds true for any $n > 0$.
Next, you need simply show that $n$ and $m$ distinct implies $w_n$ does not equal $w_m$ and you're done (Hint: you'll probably use the other fact that we haven't used yet).
1
Note $xw_n = 1$ so you need to show that the $w_n$ give infinitely many different elements, that is, $w_n = w_m$ implies $n = m$.
Hint: Assume $w_n = w_m$ and $n \neq m$. Show that $x$ has a left inverse, it also has a right inverse ($y$). Finally, show that if an element has a left and a right inverse then these two elements are equal. But this contradicts $yx \neq 1$ so our assumption that $n \neq m$ was incorrect.
Jim
- 30,682