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In this paper we recall the construction of scalar field action on$\kappa$-Minkowski space-time and investigate its properties. In particular weshow how the co-product of $\kappa$-Poincar\'e algebra of symmetries arisesfrom the analysis of the symmetries of the action, expressed in terms ofFourier transformed fields. We also derive the action on commuting space-time,equivalent to the original one. Adding the self-interaction $\Phi^4$ term weinvestigate the modified conservation laws. We show that the local interactionson $\kappa$-Minkowski space-time give rise to 6 inequivalent ways in whichenergy and momentum can be conserved at four-point vertex. We discuss therelevance of these results for Doubly Special Relativity.Comment: 17 pages; some editing done, final version to be published in Int. J.  Mod. Phys.

SCALAR FIELD THEORY ON κ-MINKOWSKI SPACE–TIME AND DOUBLY SPECIAL RELATIVITY