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Direct sum of abelian groups


The direct sum is an operation from abstract algebra, a branch of mathematics. For example, the direct sum , where is real coordinate space, is the Cartesian plane, . To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as rings, modules, and vector spaces.


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