2-sided ideal in $\mathfrak{M}_{2\times 2}(\mathbb{Z})$ ring

Question

Hi there: I know $\mathfrak{M}_{2\times 2}(\mathbb{Z})$ has left and right ideal, but is it true it does not have 2-sided ideal? If there is, could you give me an example. Thank you very much.

Answer 1

It certainly has many two sided ideals, for instance the set of all matrices whose coefficients are multiples of a given integer $d$ (i.e. $d\cdot M_{2}(\Bbb Z)$) form an ideal for every $d\in\Bbb N$.

It is a general fact (alluded to by Quimey's answer) that the two-sided ideals of a matrix ring $M_n(R)$) are in one to one correspondence with the two-sided ideals of the ring $R$ itself, in that every two-sided ideal of $M_n(R)$ is of the form $M_n(I)$ where $I$ is a two-sided ideal of $R$.

Answer 2

Yes, there are two sided ideals. For example $$ \{2 M:\text{$M$ a $2\times 2$ matrix}\} $$ In general there is a correspondence between the two-sided ideals of a ring of matrices and the two-sided ideals of the base ring, see wikipedia.

Answer 3

If $0 \ne n \in \Bbb Z$, and $(n)$ is the principal ideal generated by $(n)$, then $\mathfrak M_{2 \times 2}((n))$ is a two-sided ideal in $\mathfrak M_{2 \times 2}(\Bbb Z)$.

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!