Unit ideal in non-commutative rings - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-23T12:30:33Z http://mathoverflow.net/feeds/question/47534 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47534/unit-ideal-in-non-commutative-rings Unit ideal in non-commutative rings ashpool 2010-11-27T20:35:14Z 2010-11-27T21:13:01Z <p>In a non-commutative ring (with identity), is it possible for an element which does not possess left or right inverses to generate the entire ring? i.e. $(r)=R$, where (r) is the two-sided ideal generated by $r\in R$ ?</p> http://mathoverflow.net/questions/47534/unit-ideal-in-non-commutative-rings/47540#47540 Answer by Jan Weidner for Unit ideal in non-commutative rings Jan Weidner 2010-11-27T21:05:21Z 2010-11-27T21:13:01Z <p>Take for example the endomorphism ring of a finite dimensional vector space. There are only trivial two sided ideals. So the ideal generated by a non zero element is the whole ring.</p> http://mathoverflow.net/questions/47534/unit-ideal-in-non-commutative-rings/47541#47541 Answer by Tim Dokchitser for Unit ideal in non-commutative rings Tim Dokchitser 2010-11-27T21:07:36Z 2010-11-27T21:07:36Z <p>Yes, for example $r=\begin{pmatrix}1&amp;0\cr0&amp;0\end{pmatrix}$ generates the whole 2x2 matrix ring, $$\begin{pmatrix}1&amp;0\cr0&amp;0\end{pmatrix} + \begin{pmatrix}0&amp;0\cr1&amp;0\end{pmatrix} \cdot \begin{pmatrix}1&amp;0\cr0&amp;0\end{pmatrix} \cdot \begin{pmatrix}0&amp;1\cr0&amp;0\end{pmatrix} = \begin{pmatrix}1&amp;0\cr0&amp;1\end{pmatrix}.$$</p>