Абстракт алгебрадағы қосымша операциясы: Тікелей қосынды

Operation in abstract algebra composing objects into "more complicated" objects

The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. As an example, 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. For example, the direct sum , where is real coordinate space, is the Cartesian plane, A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, for example , provided and are the same kinds of algebraic structures (e. g., all abelian groups, or all vector spaces). This relies on the fact that the direct sum is associative up to isomorphism. That is, for any algebraic structures , , and of the same kind. The direct sum is also commutative up to isomorphism, i. e. for any algebraic structures and of the same kind. The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups. In the case where infinitely many objects are combined, the direct sum and direct product are not isomorphic, even for abelian groups, vector spaces, or modules. As an example, consider the direct sum and direct product of (countably) infinitely many copies of the integers. An element in the direct product is an infinite sequence, such as (1,2,3, ) but in the direct sum, there is 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. Often, 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 strictly smaller when the index set is infinite, because an element of the direct product can have infinitely many nonzero coordinates.

The xy plane, a two dimensional vector space, can be thought of as the direct sum of two one dimensional vector spaces, namely the x and y axes. In this direct sum, the x and y axes intersect only at the origin (the zero vector). Addition is defined coordinate wise, that is , which is the same as vector addition. Given two structures and , their direct sum is written as Given an indexed family of structures , indexed with , the direct sum may be written Each Ai is called a direct summand of A. If the index set is finite, the direct sum is the same as the direct product. In the case of groups, if the group operation is written as the phrase "direct sum" is used, while if the group operation is written the phrase "direct product" is used. When the index set is infinite, the direct sum is not the same as the direct product since the direct sum has the extra requirement that all but finitely many coordinates must be zero.

A distinction is made between internal and external direct sums, though the two are isomorphic. If the summands are defined first, and then the direct sum is defined in terms of the summands, we have an external direct sum. For example, if we define the real numbers and then define the direct sum is said to be external. If, on the other hand, we first define some algebraic structure and then write as a direct sum of two substructures and , then the direct sum is said to be internal. In this case, each element of is expressible uniquely as an algebraic combination of an element of and an element of For an example of an internal direct sum, consider (the integers modulo six), whose elements are This is expressible as an internal direct sum .

The direct sum of modules is a construction which combines several modules into a new module. The most familiar examples of this construction occur when considering vector spaces, which are modules over a field. The construction may also be extended to Banach spaces and Hilbert spaces.

However, the direct sum (defined identically to the direct sum of abelian groups) is not a coproduct of the groups and in the category of groups. So for this category, a categorical direct sum is often simply called a coproduct to avoid any possible confusion.

Some authors will speak of the direct sum of two rings when they mean the direct product , but this should be avoided since does not receive natural ring homomorphisms from and : in particular, the map sending to is not a ring homomorphism since it fails to send 1 to (assuming that in ). Thus is not a coproduct in the category of rings, and should not be written as a direct sum. (The coproduct in the category of commutative rings is the tensor product of rings. In the category of rings, the coproduct is given by a construction similar to the free product of groups.) Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If is an infinite collection of nontrivial rings, then the direct sum of the underlying additive groups can be equipped with termwise multiplication, but this produces a rng, that is, a ring without a multiplicative identity.

For any arbitrary matrices and , the direct sum is defined as the block diagonal matrix of and if both are square matrices (and to an analogous block matrix, if not).

The direct sum comes equipped with a projection homomorphism for each j in I and a coprojection for each j in I. Given another algebraic structure (with the same additional structure) and homomorphisms for every j in I, there is a unique homomorphism , called the sum of the gj, such that for all j. Thus the direct sum is the coproduct in the appropriate category.