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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?

  • 0
    What do you mean by (2)? What would be this integer in a general $\mathbb Z$-algebra?2012-07-24
  • 0
    To clarify, we can rephrase the second property as: whenever I is a two-sided ideal of R, then I = nR for some n in Z2012-07-24
  • 2
    Wait, $M_n(\mathbb Z)$ is a *free* $\mathbb Z$-algebra? Don't the matrices have nontrivial relations though?2012-07-25

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