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We study presentations of the virtual dualizing modules of special linear groups of number rings, the Steinberg modules. Bykovskiȋ gave a presentation for the Steinberg modules of the integers, and our main result is a generalization of this to the Gaussian integers and the Eisenstein integers. We also show that this generalization does not give a presentation for the Steinberg modules of several Euclidean number rings.

We prove that the general linear groups of the integers, Gaussian integers, and Eisenstein integers satisfy homological stability of slope 1 when using Z[ 1 / 2 ]-coefficients and of slope 2 / 3 when using Z-coefficients.

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