The molten magma does solidifies at the center of the ridge at the seafloor spreading.
Seafloor spreading occurs when the Divergent plates continues to move apart, thus, creating a new seafloor.
Usually, a new crust (oceanic crust) are created by the Molten magma which pushes up from the mantle.
A very good example is the Mid-Atlantic Ridge
Therefore, the molten magma does solidifies at the center of the ridge at the seafloor spreading.
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