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I am interested in rings $R$ with the following properties:

(1) $R$ is a free $\mathbb{Z}$-algebra of finite rank

(2) each two-sided ideal of $R$ is generated by an integer

The matrix rings $M_n(\mathbb{Z})$ satisfy these properties. Does anyone know of any other examples, or do these properties characterize the matrix rings over the integers?

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    Wait, $M_n(\mathbb Z)$ is a *free* $\mathbb Z$-algebra? Don't the matrices have nontrivial relations though?2012-07-25

1 Answers 1

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$M_n(\mathbb{Z})$ is not the free ring on any number of generators unless $n = 1$, where it is the free ring on no generators. This is because the free ring on a nonzero number of generators is infinitely generated as an abelian group (while $M_n(\mathbb{Z})$ is a free abelian group on $n^2$ generators): for example, the free ring on one generator is $\mathbb{Z}[x]$.

In fact these two properties uniquely characterize $\mathbb{Z}$. It's clear that the free rings on a nonzero number of generators have many two-sided ideals, because they have many quotients (corresponding to rings with that same number of generators).