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. 
@article{FPM_2012_17_7_a6,
     author = {A. L. Kanunnikov},
     title = {Orthogonal graded completion of graded semiprime rings},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {117--150},
     publisher = {mathdoc},
     volume = {17},
     number = {7},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2012_17_7_a6/}
}
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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/