The mixing of a passive scalar like lithium, beryllium or temperature fluctuations due to the magnetic Tayler instability of a rotating axial pinch is considered. Our study is carried out within a Taylor-Couette setup for two rotation laws: quasi-Kepler and solid-body rotation. The minimum magnetic Prandtl number used is 0.05 while the molecular Schmidt number Sc of the fluid varies between 0.1 and 2. An effective diffusivity coefficient for the mixing is numerically measured by the decay process of a global concentration peak located between the cylinder walls. We find that only models with Sc>0.1 do provide finite eddy diffusivity values. We also find that for quasi-Kepler rotation at a magnetic Mach number Mm~2 the flow transits from the slow-rotation regime to the fast-rotation regime. For fixed Reynolds number the relation between the normalized eddy diffusivity and the Schmidt number of the fluid is always linear so that also a linear relation between the instability-induced diffusivity and the molecular viscosity results just in the sense proposed by Schatzman (1977). The numerical value of the coefficient in this relation will reach a maximum at Mm~2 and will decrease for Mm>>1 implying that only toroidal magnetic fields of order kG can exist in the solar tachocline.
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