Abstract
A module over a semiring lacks zero sums (LZS) if it has the property thatv + w = 0 implies v = 0 and w = 0. While modules over a ring never lack zero sums, thisproperty always holds for modules over an idempotent semiring and related semirings, soarises for example in tropical mathematics.A direct sum decomposition theory is developed for direct summands (and complements)of LZS modules: The direct complement is unique, and the decomposition is unique up torefinement. Thus, every finitely generated “strongly projective” module is a finite directsum of summands of R (assuming the mild assumption that 1 is a finite sum of orthogonalprimitive idempotents of R). This leads to an analog of the socle of “upper bound”modules. Some of the results are presented more generally for weak complements and semicomplements.We conclude by examining the obstruction to the “upper bound” propertyin this context.
Original language  English 

Pages (fromto)  503524 
Number of pages  22 
Journal  Israel Journal of Mathematics 
Volume  225 
Issue number  2 
Early online date  11 Apr 2018 
DOIs  
Publication status  Published  30 Apr 2018 
Keywords
 semiring
 lacking zero sums
 direct sum decomposition
 free (semi)module
 projective (semi)module
 indecomposable
 semicomplement
 weak complement
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Profiles

Zur Izhakian
 School of Natural & Computing Sciences, Mathematical Science  Reader
 Mathematical Sciences (Research Theme)
Person: Academic