Geodesic
Orthogonal graded completion of graded semiprime rings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 7, pp. 117-150.

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For an associative $\mathrm{gr}$-semiprime ring $R$ with identity graded by a group, the orthogonal graded completion $O^\mathrm{gr}(R)$ is constructed. A criterion for the orthogonal completeness of the maximal right graded quotient ring $Q^\mathrm{gr}(R)$ is proved. The ring $Q^\mathrm{gr}(R)$ need not be orthogonally complete, as opposed to the ungraded case. 
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A. L. Kanunnikov. Orthogonal graded completion of graded semiprime rings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 7, pp. 117-150. http://geodesic.mathdoc.fr/item/FPM_2012_17_7_a6/

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