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- Direct_sum abstract "The direct sum is an operation from abstract algebra, a branch of mathematics. As an example, consider the direct sum , where is the set of real numbers. is the Cartesian plane, the xy-plane from elementary algebra. In general, the direct sum of two objects is another object of the same type, so the direct sum of two geometric objects is a geometric object and the direct sum of two sets is a set.To see how direct sum is used in abstract algebra, consider the most 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.We can also form direct sums with any number of summands, for example , provided and are the same kinds of algebraic structures, that is, all groups or all rings or all vector spaces. In the case of two summands, or any finite number of summands, the direct sum is the same as the direct product. If the arithmetic operation is written as +, as it usually is in abelian groups, then we use the direct sum. If the arithmetic operation is written as × or ⋅ or using juxtaposition (as in the expression ) we use direct product.In the case where infinitely many objects are combined, most authors make a distinction between direct sum and direct product. As an example, consider the direct sum and direct product of infinitely many real lines. An element in the direct sum is an infinite sequence, such as (1,2,3,...) but in the direct sum, there would be a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. More generally, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are , the direct sum is defined to be the set of tuples with such that for all but finitely many i. The direct sum is contained in the direct product , but is usually strictly smaller when the index set is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero.".
- Direct_sum wikiPageID "2084687".
- Direct_sum wikiPageRevisionID "603499745".
- Direct_sum hasPhotoCollection Direct_sum.
- Direct_sum subject Category:Abstract_algebra.
- Direct_sum comment "The direct sum is an operation from abstract algebra, a branch of mathematics. As an example, consider the direct sum , where is the set of real numbers. is the Cartesian plane, the xy-plane from elementary algebra.".
- Direct_sum label "Direct sum".
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- Direct_sum sameAs Q13582243.
- Direct_sum sameAs Q13582243.
- Direct_sum wasDerivedFrom Direct_sum?oldid=603499745.
- Direct_sum isPrimaryTopicOf Direct_sum.