Geodesic
A note on orthodox additive inverse semirings
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 43 (2004) no. 1, pp. 149-154.

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We show in an additive inverse regular semiring $(S, +, \cdot )$ with $E^{\bullet }(S)$ as the set of all multiplicative idempotents and $E^+(S)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e, f \in E^{\bullet }(S)$, $ef \in E^+(S)$ implies $fe\in E^+(S)$. (ii) $(S, \cdot )$ is orthodox. (iii) $(S, \cdot )$ is a semilattice of groups. This result generalizes the corresponding result of regular ring.
Classification : 16A78, 16E50, 16Y60, 20M07, 20M10
Keywords: additive inverse semirings; regular semirings; orthodox semirings 
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Sen, M. K.; Maity, S. K. A note on orthodox additive inverse semirings. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 43 (2004) no. 1, pp. 149-154. http://geodesic.mathdoc.fr/item/AUPO_2004__43_1_a15/