Geodesic
Semirings of cyclic types
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 2, pp. 175-192.

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The authors investigate semirings of cyclic types from the algebraic point of view. To simplify and facilitate the analysis, local Fourier transform of these semirings is introduced. The authors describe zero divisors, nilpotent elements, invertible elements, idempotents, and the Jacobson radical. 
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A. V. Mikhalev; A. A. Nechaev; A. V. Prudnikov; M. S. Staroverov; A. S. Vydrin. Semirings of cyclic types. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 2, pp. 175-192. http://geodesic.mathdoc.fr/item/FPM_2006_12_2_a11/

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[2] Kuzmin A. S., Kurakin V. L., Mikhalev A. V., Nechaev A. A., “Linear recurring sequences over rings and modules”, J. Math. Sci., 76:6 (1995), 2793–2915 | DOI | MR | Zbl