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In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
The pushout of these maps is the direct sum of A and B. Generalizing to the case where f and g are arbitrary homomorphisms from a common domain Z, one obtains for the pushout a quotient group of the direct sum; namely, we mod out by the subgroup consisting of pairs (f(z), −g(z)). Thus we have "glued" along the images of Z under f and g.
Proofs often become short and elegant if the universal property is used rather than the concrete details. For example, the tensor algebra of a vector space is slightly complicated to construct, but much easier to deal with by its universal property. Universal properties define objects uniquely up to a unique isomorphism. [1]
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.
The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected (and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ. This fails if J is not connected. For example, if J is a discrete category, the components of the unit are the diagonal morphisms ...
For example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms. (It therefore coincides ...
M is a direct sum of countably generated modules, M satisfies a certain Mittag-Leffler -type condition. This characterization can be used to show that if R → S {\displaystyle R\to S} is a faithfully flat map of commutative rings and M {\displaystyle M} is an R {\displaystyle R} -module, then M {\displaystyle M} is projective if and only if M ...
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