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## Main Question or Discussion Point

I read the following on the wikipedia page about simple rings (http://en.wikipedia.org/wiki/Simple_ring): [Broken]

[tex]

\begin{pmatrix}

0 &1 &1\\

0 &0 &0\\

0 &0 &0

\end{pmatrix}

[/tex]

then J is not equal to S = {M ∈ M(3,ℝ) | The 1st column of M has zero entries},

since for example the following matrix is in S but not in J:

[tex]

\begin{pmatrix}

0 &1 &0\\

0 &0 &1\\

0 &0 &0

\end{pmatrix}

[/tex]

Can anybody help me understand what the wikipedia page is trying to say, or where I am seeing things wrong?

I do not see why this is the case. Take the ring of 3 by 3 matrices over the real numbers and the left ideal, J, generated by:Let D be a division ring and M(n,D) be the ring of matrices with entries in D. It is not hard to show that every left ideal in M(n,D) takes the following form:

{M ∈ M(n,D) | The n1...nk-th columns of M have zero entries},

for some fixed {n1,...,nk} ⊂ {1, ..., n}.

[tex]

\begin{pmatrix}

0 &1 &1\\

0 &0 &0\\

0 &0 &0

\end{pmatrix}

[/tex]

then J is not equal to S = {M ∈ M(3,ℝ) | The 1st column of M has zero entries},

since for example the following matrix is in S but not in J:

[tex]

\begin{pmatrix}

0 &1 &0\\

0 &0 &1\\

0 &0 &0

\end{pmatrix}

[/tex]

Can anybody help me understand what the wikipedia page is trying to say, or where I am seeing things wrong?

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