The direct sum of two Modules and over the same Ring is given by
with Module operations defined by
The dimension of a direct sum is the product of the dimensions of the quantities summed. The significant property of the direct sum is that it is the coproduct in the category of Modules. This general definition gives as a consequence the definition of the direct sum of Abelian Groups (since they are Modules over the Integers) and the direct sum of Vector Spaces (since they are Modules over a Field).