The forced and natural flows of fluid within an annulus caused by the rotation of cylinders and temperature differences of the inner and outer walls are observed in various engineering applications. In this research, the laminar flow regime and mixed convection inside a ring-shaped horizontal concentric and eccentric space for an incompressible fluid are studied in the existence of an axial magnetic field. The present work is the first effort to investigate the influence of a magnetic field on flow and combined-convection heat exchange characteristics within an annulus with a cold outer cylinder and an inner hot cylinder. Here, the properties of the flow and heat transfer characteristics are studied using the finite volume method. Numerical procedures are mainly investigated for recognizing the influence of Hartmann number (in the range of 0 ≤ Ha ≤ 100), as the representative of the magnetic force, on velocity components, Nusselt number, streamlines, and isothermal lines. One of the notable effects is that when Ha number increases, it will reduce the vorticity of the fluid and buoyancy forces. As a result, streamlines and isothermal lines can be seen more constant as regular concentric circles. A rise in Ha number decreases the range of local Nu number variation for both cylinders. The average Nu number for the outer and inner cylinders has different trends when Ha number increases. Taking concentric cylinders as an example, this parameter for the inner and the outer cylinders increases and decreases by about 1.2 and 1.6, respectively.