Geodesic
Duo Rings: Some Applications to Commutativity Theorems
Canadian mathematical bulletin, Tome 11 (1968) no. 3, pp. 375-380

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Proofs of commutativity theorems for general rings usually employ the Jacobson structure theory; however, alternative approaches to the "xn = x theorem" [ l, 2] suggest that the power of the Jacobson theory is not required. In this note we prove two commutativity theorems of Herstein in an elementary way. Both proofs involve establishing first that the rings under consideration are duo-rings - rings in which every one-sided ideal is two-sided. 
Bell, Howard E. Duo Rings: Some Applications to Commutativity Theorems. Canadian mathematical bulletin, Tome 11 (1968) no. 3, pp. 375-380. doi: 10.4153/CMB-1968-042-0
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     author = {Bell, Howard E.},
     title = {Duo {Rings:} {Some} {Applications} to {Commutativity} {Theorems}},
     journal = {Canadian mathematical bulletin},
     pages = {375--380},
     year = {1968},
     volume = {11},
     number = {3},
     doi = {10.4153/CMB-1968-042-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-042-0/}
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