Geodesic
Are One-Sided Inverses Two-Sided Inverses in a Matrix Ring Over a Group Ring?
Canadian mathematical bulletin, Tome 13 (1970) no. 4, pp. 475-479

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A ring R with identity element is n-finite if for any pair A, B of n × n matrices over R, AB = In implies BA = In . In module theoretic terms, R is n-finite if and only if in a free R-module of rank n any generating set of n elements is free. If R is n-finite for all positive integers n then R is said to be strongly finite. It is known that all commutative rings, all Artinian rings and all Noetherian rings are strongly finite. These and many other interesting results appear in a paper of P. M. Cohn [1]. 
Losey, Gerald. Are One-Sided Inverses Two-Sided Inverses in a Matrix Ring Over a Group Ring?. Canadian mathematical bulletin, Tome 13 (1970) no. 4, pp. 475-479. doi: 10.4153/CMB-1970-087-3
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     author = {Losey, Gerald},
     title = {Are {One-Sided} {Inverses} {Two-Sided} {Inverses} in a {Matrix} {Ring} {Over} a {Group} {Ring?}},
     journal = {Canadian mathematical bulletin},
     pages = {475--479},
     year = {1970},
     volume = {13},
     number = {4},
     doi = {10.4153/CMB-1970-087-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1970-087-3/}
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