Show that R is a skew field.

Question

Let $R$ be a ring with unity such that the only left ideals of $R$ are $\{0\}$ and $R$. Show that $R$ is a skew field.

Please help me to solve this problem.

If $x\neq0$ what can you say about the left ideal it generates? — Jyrki Lahtonen, Dec 09 '16 at 14:50
The definition of skew field might be a good start. Do you know that? — Matthew Leingang, Dec 09 '16 at 15:00
@MatthewLeingang I know that in skew field every non-zero elements is a unit. — user1942348, Dec 09 '16 at 15:05

Learnmore · Accepted Answer · 2016-12-09T15:02:22.523

2

Let $a\neq 0$ .

Consider $Ra$ .(Show that it is a left ideal of $R$).

$a=1.a\in Ra\implies Ra\neq \{0\}$.

Conclusion $Ra=R\implies \exists b\in R$ such that $ba=1$.

To conclude the proof show that $ab=ba=1$(How?)

Also similarly $Rb=R\implies cb=1 $ for some $c\in R$.Now $a=1.a=(cb).a=c.(ba)=c.1=c$

edited Dec 09 '16 at 15:02

answered Dec 09 '16 at 14:52

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how you have concluded Ra=R – user1942348 Dec 09 '16 at 14:55
thanks. it is correct. i know that in skew field every non-zero elements is a unit. have you used this definition to prove skew field? How ba=1 imply ab=1 – user1942348 Dec 09 '16 at 14:59
@user1942348 Yes, that's what he has done. He's proven that any non-zero element generates a left ideal with $1$ in it, and therefore must have an inverse. – Arthur Dec 09 '16 at 15:00
Bcs $Ra$ is a left ideal and only left ideals are $R$ and ${0}$ – Learnmore Dec 09 '16 at 15:03
@user1942348;see my edits by hovering you mouse over shaded part – Learnmore Dec 09 '16 at 15:05

1 Answers1