Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Assume that $P$ is a ring and $M_n(P)$ is a ring of matrices over $P$. If $P$ is an unitary ring then every two-sided ideal in $M_n(P)$ is of the form $M_n(I)$, where $I$ is a two-sided ideal in $P$ (see for example here).

Is the same true for non-unitary rings?


share|cite|improve this question
up vote 3 down vote accepted

No. For example, take the ring of $2\times 2$ matrices over the even integers, $2\mathbb{Z}$. Let $J$ be the subring of all matrices of the form $$\left(\begin{array}{cc} 2a & 4b\\ 4c & 4d \end{array}\right)$$ with $a,b,c,d\in\mathbb{Z}$. This is an ideal, since it is plainly a subgroup, and $$\begin{align*} \left(\begin{array}{cc} 2\alpha & 2\beta\\ 2\gamma & 2\delta\end{array}\right)\left(\begin{array}{cc} 2a & 4b\\ 4c & 4d \end{array}\right) &= \left(\begin{array}{cc} 4(\alpha a + 2\beta c) & 8(\alpha b + \beta d)\\ 4(\gamma a + 2\delta c) & 8(\gamma b + \delta d) \end{array}\right)\in J,\\ \left(\begin{array}{cc} 2a & 4b\\ 4c & 4d \end{array}\right)\left(\begin{array}{cc} 2\alpha & 2\beta\\ 2\gamma & 2\delta \end{array}\right) &= \left(\begin{array}{cc} 4(\alpha a+2\gamma b) & 4(\beta a + 2 \delta b)\\ 8(\alpha c + \gamma d) & 8(\beta c + \delta d) \end{array}\right)\in J. \end{align*}$$ However, $J$ is not of the form $M_2(I)$ for an ideal $I$ of $2\mathbb{Z}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.