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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 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.
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 ...
So M breaks up as the direct sum of R-modules, M = e 1 M ⊕ ... ⊕ e n M. Conversely, given an R-module M 0, then M 0 ⊕n is an M n (R)-module. In fact, the category of R-modules and the category of M n (R)-modules are equivalent. The special case is that the module M is just R as a module over itself, then R n is an M n (R)-module.
A decomposition with local endomorphism rings [5] (cf. #Azumaya's theorem): a direct sum of modules whose endomorphism rings are local rings (a ring is local if for each element x, either x or 1 − x is a unit). Serial decomposition: a direct sum of uniserial modules (a module is uniserial if the lattice of submodules is a finite chain [6]).
Group cohomology theory also has a direct application in condensed matter physics. Just like group theory being the mathematical foundation of spontaneous symmetry breaking phases, group cohomology theory is the mathematical foundation of a class of quantum states of matter—short-range entangled states with symmetry.
Shield AI, a defense tech startup valued at $2.8 billion, has agreed to buy Sentient Vision Systems for an undisclosed sum.. Melbourne-based Sentient has developed AI-enabled edge solutions that ...
The unit of this adjoint pair is the defining pair of inclusion maps from X 1 and X 2 into the direct sum, and the counit is the additive map from the direct sum of (X,X) to back to X (sending an element (a,b) of the direct sum to the element a+b of X).