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...

can be used to form the direct sum of two vector spaces or two modules. Direct sums can also be formed with any finite number of summands; for example,...

groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information. A group...

decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for...

the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific"...

up direct in Wiktionary, the free dictionary. Direct may refer to: Directed set, in order theory Direct limit of (pre), sheaves Direct sum of modules, a...

objects Direct sum of groups Direct sum of modules Direct sum of permutations Direct sum of topological groups Einstein summation, a way of contracting...

factors are the whole space is the box topology. The elements of the direct sum of modules ⨁ M i {\displaystyle \bigoplus M_{i}} are sequences α i ∈ M i...

of irreducible modules. M is the sum of its irreducible submodules. Every submodule of M is a direct summand: for every submodule N of M, there is a complement...

coproduct coincide for finite collections of objects. The biproduct is a generalization of finite direct sums of modules. Let C be a category with zero morphisms...

every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module over one ring may...

semisimple, which is a direct sum of simple modules. A direct sum decomposition of a module into indecomposable modules is called an indecomposable decomposition...

irreducible. Semisimple A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible...

} denote the direct sum of modules. The second syzygy module is the module of the relations between generators of the first syzygy module. By continuing...

anti-symmetrizing, so the algebra decomposes as a direct sum of modules (vector spaces if the *-ring is a field) of symmetric and anti-symmetric (Hermitian and...

decomposition into a direct sum of copies of R (rank one free modules) is replaced by a direct sum into rank one projective modules: the individual summands...

compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations...

class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free...

1958), which states that a projective module is a direct sum of countably generated modules. More generally, a module over a possibly non-commutative ring...

category R-Mod of (left) modules over a ring R (commutative or not) becomes a cocartesian monoidal category with the direct sum of modules as tensor product...

structure). Given a module M over a commutative ring R, the direct sum of modules R ⊕ M has a structure of an R-algebra by thinking M consists of infinitesimal...

of positive roots. Algebraic characters are defined for locally-finite weight modules and are additive, i.e. the character of a direct sum of modules...

product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. The direct sum is...

Complemented subspace Direct sum – Operation in abstract algebra composing objects into "more complicated" objects Direct sum of modules – Operation in abstract...

direct sum of modules over a ring. Example: A typical application of the Eilenberg swindle in algebra is the proof that if A is a projective module over...

cartesian product of n copies of R as a left R-module, is free. If R has invariant basis number, then its rank is n. A direct sum of free modules is free, while...

over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent...

category of modules over R {\displaystyle R} . (One can take this to mean either left R {\displaystyle R} -modules or right R {\displaystyle R} -modules.) For...

objects into "more complicated" objects Direct sum of modules – Operation in abstract algebra Direct sum of topological groups Y {\displaystyle Y} is...

{\displaystyle N_{2}\subseteq N_{1}} . A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial...