Cofiniteness - Other Examples - Direct Sum

Direct Sum

The elements of the direct sum of modules are sequences where cofinitely many .

The analog (without requiring that cofinitely many are zero) is the direct product.

Read more about this topic:  Cofiniteness, Other Examples

Other articles related to "direct sum, direct":

Projection (linear Algebra) - Projections On Normed Vector Spaces
... A given direct sum decomposition of X into complementary subspaces still specifies a projection, and vice versa ... If X is the direct sum X = U ⊕ V, then the operator defined by P(u + v) = u is still a projection with range U and kernel V ... The relation I = P + (I − P) implies X is the direct sum Ran(P) ⊕ Ran(I − P) ...
... into 'simple' ('irreducible') parts, that fit together in the cleanest way (by direct sum) ... Briefly, "semisimple = direct sum of simple", or equivalently "completely reducible = direct sum of irreducible" ... module is one in which each submodule is a direct summand ...
Maschke's Theorem - Reformulation and The Meaning
... representation built from irreducible subrepresentations using the direct sum operation? In the module-theoretic language, is an arbitrary module semisimple? In ... since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents) ... theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities ...
Internal Direct Sum
... An internal direct sum is simply a direct sum of subobjects of an object ... For example, the real vector space R2 = {(x, y) x, y ∈ R} is the direct sum of the x-axis {(x, 0) x ∈ R} and the y-axis {(0, y) y ∈ R}, and ... More generally, given a vector space V and two subspaces U and W, V is the (internal) direct sum U ⊕ W if U + W = {u + w u ∈ U, w ∈ W} = V, and if u + w = 0 with u ∈ U and w ∈ W, then u = w = 0 ...
Split-biquaternion - Direct Sum of Two Quaternion Rings
... The direct sum of the division ring of quaternions with itself is denoted ... The product of two elements and is in this direct sum algebra ... on the basis of the Proposition it is apparent that Clifford biquaternions split into the direct sum of real quaternions ...

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