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In this paper we discuss various correlations measured by the concurrence(C), classical correlation (CC), quantum discord (QD), and geometric measure ofdiscord (GMD) in a two-qubit Heisenberg XXZ spin chain in the presence ofexternal magnetic field and Dzyaloshinskii-Moriya (DM) anisotropicantisymmetric interaction. Based on the analytically derived expressions forthe correlations for the cases of thermal equilibrium and the inclusion ofintrinsic decoherence, we discuss and compare the effects of various systemparameters on the correlations in different cases. The results show that theanisotropy Jz is considerably crucial for the correlations in thermalequilibrium at zero temperature limit but ineffective under the considerationof the intrinsic decoherence, and these quantities decrease as temperature Trises on the whole. Besides, J turned out to be constructive, but B bedetrimental in the manipulation and control of various quantities both inthermal equilibrium and under the intrinsic decoherence which can be avoided bytuning other system parameters, while D is constructive in thermal equilibrium,but destructive in the case of intrinsic decoherence in general. In addition,for the initial state $|\Psi_1(0) > = \frac{1}{\sqrt{2}} (|01 > + |10 >)$, allthe correlations except the CC, exhibit a damping oscillation to a stable valuelarger than zero following the time, while for the initial state $|\Psi_2(0) >= \frac{1}{\sqrt{2}} (|00 > + |11 >)$, all the correlations monotonouslydecrease, but CC still remains maximum. Moreover, there is not a definiteordering of these quantities in thermal equilibrium, whereas there is adescending order of the CC, C, GMD and QD under the intrinsic decoherence witha nonnull B when the initial state is $|\Psi_2(0) >$.

Various correlations in a Heisenberg XXZ spin chain both in thermal equilibrium and under the intrinsic decoherence