Primary decomposition of A-ideals in MV-semimodules

Document Type : Original Article


Department of Mathematics, Farhangian University, Tehran, Iran


In [16], by using an MV-semiring and an MV-algebra, we introduced the new definition  of MV-semimodule and studied some of their basic properties. In this paper, we study  and present definitions of primary ideals of MV-semirings, decomposition of ideals in  MV-semirings, primary A-ideals of MV -semimodules, and decomposition of A-ideals in MV-semimodules. Then we present some conditions that an A-ideal can have a reduced primary decomposition. 


[1] L.P. Belluce, A. Di Nola, Commutative rings whose ideals form an MV-algebra, Mathematical
Logic Quarterly, 55(5) (2009), 468–486.
[2] R.A. Borzooei, S. Saidi Goraghani, Free MV-modules, Journal of Intelligent and Fuzzy Systems,
31(1) (2016), 151–161.
[3] C.C. Chang, Algebraic analysis of many-valued logic, Transactions of the American Mathematical
Society, 88 (1958), 467–490.
[4] C.C. Chang, A new proof of the completeness of the ukasiewicz axioms, Transactions of the
American Mathematical Society, 93 (1959), 74–80.
[5] R. Cignoli, I.M.L. D, Ottaviano, D. Mundici, Algebric foundation of many-valued reasoning,
Kluwer Academic, Dordrecht, 2000.
[6] A. Di Nola, P. Flondor, I. Leustean, MV -modules, Journal of Algebra, 267 (2003), 21–40.
[7] A. Di Nola, B. Gerla, Algebras of Łukasiewicz’s logic and their semiring reducts, Contemporary
Mathematics, 377 (2005), 131–144.
[8] A. Di Nola, C. Russo, Semiring and semimodules issues in MV -algebras, Communications in
Algebra, 41(3) (2013), 1017–1048.
[9] F. Forouzesh, E. Eslami, A. Borumand Saeid, Radical of A-ideals in MV -modules, Analele
Stiintifice ale Universitatii Al l Cuza din Iasi-Mathematica, LXII (f. 1) (2016), 33–57.
[10] J. S. Golan, Semirings and their applications, Kluwer Academic Publishers, Dordrecht, 1999.
[11] L. Peter Belluce, A. Di Nola, A. Rita Ferraioli, MV-semirings and their sheaf representations,
Order, 30 (2013), 165–179.
[12] S. Saidi Goraghani, R.A. Borzooei, Most results on A-ideals in MV-moduls, Journal of Algebraic
Systems, 5(1) (2017), 1–13.
[13] S. Saidi Goraghani, R.A. Borzooei, Results on prime ideals in PMV -algebras and MV -
modules, Italian Journal of Pure and Applied Mathematics, 37 (2017), 183–196.
[14] S. Saidi Goraghani, R.A. Borzooei, Decomposition of A-ideals in MV-modules, Annals of the
University of Craiova-mathematics and Computer Science Series, 45(1) (2018), 66–77.
[15] S. Saidi Goraghani, R.A. Borzooei, On injective MV -modules, Bulletin of of the Section of
Logic, 47(4) (2018), 283–298.
[16] S. Saidi Goraghani, R.A. Borzooei, S.S. Ann, New definition of MV-semimoduls, Journal of
Intelligent and Fuzzy Systems, 41 (2021), 3715–3726.