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The Dzyaloshinskii-Moriya interaction (DMI) has explained successfully theweak ferromagnetism in some centrosymmetric antiferromagnets. However, in thelast years, it was generally claimed that the DMI is not effective incentrosymmetric systems. We reconciled these views by separating theconventional antiferromagnets and altermagnets. Altermagnets representcollinear antiferromagnetic compounds with spin-up and spin-down sublatticesconnected only by mirror and roto-translational symmetries. Consequently, thesystem shows even-parity wave spin order in the k-space lifting the Kramer'sdegeneracy in the non-relativistic band structure. We emphasize that the DMIcan create weak ferromagnetism in centrosymmetric altermagnets while it is noteffective in centrosymmetric conventional antiferromagnets. Additionally, DMIcan create weak ferromagnetism or weak ferrimagnetism in noncentrosymmetricaltermagnets. Once the spin-orbit coupling is included in an altermagneticsystem without time-reversal symmetry, the components of spin moments of thetwo sublattices along the Neel vector are antiparallel but the other two spincomponents orthogonal to the N\'eel vector can be either parallel orantiparallel for centrosymetric systems. For noncentrosymmetric systems, we canhave different bands showing parallel or antiparallel spin components resultingin weak ferrimagnetism. We can divide the altermagnetic compounds into classesbased on the weak ferromagnetism or weak ferrimagnetism properties. The weakferromagnetism and weak ferrimagnetism induced by DMI is a property exclusivelyof the altermagnets, not present in either ferromagnets or conventionalantiferromagnets so we propose that altermagnets should be classified based onthis property.

Dzyaloshinskii-Moriya interaction inducing weak ferromagnetism in  centrosymmetric altermagnets and weak ferrimagnetism in noncentrosymmetric  altermagnets