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Can a non-commutative ring $R$ contains identity?

Suppose $R$ contains the identity element 1. Construct an ideal $Z(R) = \{a \in R \mid ra = ar\text{ for all }r \in R\}$. Since $1 \in Z(R)$, $R = Z(R)$. This implies that $R$ is commutative. Contradiction.

I can't see what it wrong with my proof. However, I don't think a non-commutative ring must not contain identity. Can anyone enlighten me please?

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    $\begingroup$ The center is a subring, not an ideal. $\endgroup$ – anon Apr 19 '15 at 16:36
  • $\begingroup$ Note the proper use of \mid in my edit. $\{a\in R\mid ra=ar\text{ for all }r\in R\}$ looks different from $\{a\in R| ra=ar\text{ for all }r\in R\}$. ${}\qquad{}$ $\endgroup$ – Michael Hardy Apr 19 '15 at 17:21
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Consider the ring $M_n(\mathbb{R})$ of all $n\times n$ matrices over $\mathbb{R}$. Clearly, this is a non-commutative ring but it has an identity, namely, the $\textbf{Identity Matrix}$

Actually you don't need to compute the center explicitly. For simplicity I will exhibit the case of $n=2$, the same can be generalized for larger values of $n$.

Notice that since the Identity matrix is in $Z(M_2(\mathbb{R}))$, then for $Z(M_2(\mathbb{R}))$ to be an ideal we must have $A \in Z(M_2(\mathbb{R}))$ for any matrix $A \in M_2(\mathbb{R})$. In particular, consider the matrix \begin{equation}A = \begin{bmatrix} 0 & 0 \\[0.3em] 1 & 0 \end{bmatrix}.\end{equation} Then for the matrix $B = \begin{bmatrix} 0 & 1 \\[0.3em] 0 & 0 \end{bmatrix}$ you can see that \begin{equation}AB = \begin{bmatrix} 0 & 0 \\[0.3em] 0 & 1 \end{bmatrix} \neq BA = \begin{bmatrix} 1 & 0 \\[0.3em] 0 & 0 \end{bmatrix},\end{equation} which shows that $A$ is not in $Z(M_2(\mathbb{R}))$ and hence $Z(M_2(\mathbb{R}))$ is not an ideal.

$\textbf{Note:}$ For general $n$ pick matrices like we have picked $A$ and $B$ see that $Z(M_n(\mathbb{R}))$ is not an ideal in $M_2(\mathbb{R})$.

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    $\begingroup$ I think it would be more helpful to OP if you constructed the center of this ring and showed that it wasn't an ideal $\endgroup$ – ASKASK Apr 19 '15 at 17:08
  • $\begingroup$ @ASKASK I guess the edit will be useful. $\endgroup$ – Urban PENDU Apr 19 '15 at 20:03
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The ring of quaternions is non-commutative.

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