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Every injective module can be decomposed as direct sum of indecomposable injective modules.

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## Some Related Sentences

Every and injective

__Every__map that is

__injective__, continuous and either open or closed is an embedding ; however there are also embeddings which are neither open nor closed

**.**

__Every__morphism in a concrete category whose underlying function is

__injective__is a monomorphism ; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily

**be**a monomorphism in the categorical sense

**.**

Every and module

__Every__

__module__over a division ring has a basis ; linear maps between finite-dimensional

**modules**over a division ring

**can**

**be**described by matrices, and the Gaussian elimination algorithm remains applicable

**.**

__Every__library

__module__has at least two source files: a definitions file specifying the library's interface plus one or more program files specifying the implementation

**of**the procedures in the interface

**.**

__Every__simple R-module is isomorphic to a quotient R / m where m is a maximal right ideal

**of**R

**.**By the above paragraph, any quotient R / m is a simple

__module__

**.**

__Every__vector space is free, and the free vector space on a set is a special case

**of**a free

__module__on a set

**.**

This is up to isomorphism the only

**indecomposable**__module__over R**.**__Every__left R-module is a**direct****sum****of**( finitely or infinitely many ) copies**of**this__module__K < sup > n </ sup >.__Every__finite-length

__module__M has a composition series, and the length

**of**every such composition series is equal to the length

**of**M

**.**

__Every__ring which is semisimple

**as**a

__module__over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple

**as**a

__module__over itself

**.**

__Every__superfield, i

**.**e

**.**a field that depends on all coordinates

**of**the superspace ( or in other words, an element

**of**a

__module__

**of**the algebra

**of**functions over superspace ), may

**be**expanded with respect to the new fermionic coordinates

**.**

Every and can

__Every__such subset has a smallest element, so to specify our choice function we

__can__simply say that it maps each set to the least element

**of**that set

**.**

__Every__information exchange between living organisms — i

**.**e

**.**transmission

**of**signals that involve a living sender and receiver

__can__

**be**considered a form

**of**communication ; and even primitive creatures such

**as**corals are competent to communicate

**.**

__Every__context-sensitive grammar which does not generate the empty string

__can__

**be**transformed into an equivalent one in Kuroda normal form

**.**

__Every__grammar in Chomsky normal form is context-free, and conversely, every context-free grammar

__can__

**be**transformed into an equivalent one which is in Chomsky normal form

**.**

__Every__entire function

__can__

**be**represented

**as**a power series that converges uniformly on compact sets

**.**

Group actions / representations:

__Every__group G__can__**be**considered**as**a category with a single object whose morphisms are the elements**of**G**.**A functor from G to Set is then nothing but a group action**of**G on a particular set, i**.**e**.**a G-set**.**__Every__sequence

__can__, thus,

**be**read in three reading frames, each

**of**which will produce a different amino acid sequence ( in the given example, Gly-Lys-Pro, Gly-Asn, or Glu-Thr, respectively ).

__Every__hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree

**of**" spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it

__can__

**be**rotated so that it opens in the desired direction and

__can__

**be**translated ( rigidly moved in the plane ) so that it is centered at the origin

**.**

__Every__holomorphic function

__can__

**be**separated into its real and imaginary parts, and each

**of**these is a solution

**of**Laplace's equation on R < sup > 2 </ sup >.

__Every__species

__can__

**be**given a unique ( and, one hopes, stable ) name,

**as**compared with common names that are often neither unique nor consistent from place to place and language to language

**.**

__Every__vector v in determines a linear map from R to taking 1 to v, which

__can__

**be**thought

**of**

**as**a Lie algebra homomorphism

**.**

__Every__use

**of**modus tollens

__can__

**be**converted to a use

**of**modus ponens and one use

**of**transposition to the premise which is a material implication

**.**

__Every__adult, healthy, sane Muslim who has the financial and physical capacity to travel to Mecca and

__can__make arrangements for the care

**of**his / her dependants during the trip, must perform the Hajj once in a lifetime

**.**

*

__Every__finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood**of**y, and every finite preorder__can__**be**formed**as**the specialization preorder**of**a topological space in this way**.**
*

__Every__preorder__can__**be**given a topology, the Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set**.**__Every__binary relation R on a set S

__can__

**be**extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

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