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Every indecomposable algebraically compact module has a local endomorphism ring.

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## Some Related Sentences

Every and indecomposable

A finitely-generated abelian group is

__indecomposable__if and only if it is isomorphic to Z or to**a**factor group of the form for some prime number p and some positive integer n**.**__Every__finitely-generated abelian group is**a**direct sum of ( finitely many )__indecomposable__abelian groups**.**
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 and algebraically

__Every__field

**has**an algebraic extension which is

__algebraically__closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice

**.**

__Every__H * is very special in structure: it is pure-injective ( also called

__algebraically__

**compact**), which says more or less that solving equations in H * is relatively straightforward

**.**

__Every__algebraic plane curve

**has**

**a**degree, which can be defined, in case of an

__algebraically__closed field, as number of intersections of the curve with

**a**generic line

**.**

Every and compact

*

__Every__topological space X is**a**dense subspace of**a**__compact__space having at most one point more than X, by the Alexandroff one-point compactification**.**
*

__Every__continuous map from**a**__compact__space to**a**Hausdorff space is closed and proper ( i**.**e., the pre-image of**a**__compact__set is__compact__**.**__Every__entire function can be represented as

**a**power series that converges uniformly on

__compact__sets

**.**

*

__Every__locally__compact__regular space is completely regular, and therefore every locally__compact__Hausdorff space is Tychonoff**.**__Every__

__compact__Hausdorff space is also locally

__compact__, and many examples of

__compact__spaces may be found in the article

__compact__space

**.**

*

__Every____compact__Hausdorff space of weight at most ( see Aleph number ) is the continuous image of ( this does not need the continuum hypothesis, but is less interesting in its absence ).__Every__group

**has**

**a**presentation, and in fact many different presentations ;

**a**presentation is often the most

__compact__way of describing the structure of the group

**.**

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

**.**

__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 has

__Every__woman

__has__had the experience of saying no when she meant yes, and saying yes when she meant no

**.**

__Every__detail in his interpretation

__has__been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout

**.**

__Every__family of Riviera Presbyterian Church

__has__been asked to read the Bible and pray together daily during National Christian Family Week and to undertake one project in which all members of the family participate

**.**

__Every__community, if it is alive

__has__

**a**spirit, and that spirit is the center of its unity and identity

**.**

__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

**.**

** Zorn's lemma:

__Every__non-empty partially ordered set in which every chain ( i**.**e**.**totally ordered subset )__has__an upper bound contains at least one maximal element**.**
The restricted principle "

__Every__partially ordered set__has__**a**maximal totally ordered subset " is also equivalent to AC over ZF**.**
** Tukey's lemma:

__Every__non-empty collection of finite character__has__**a**maximal element with respect to inclusion**.**
*

__Every__continuous functor on**a**small-complete category which satisfies the appropriate solution set condition__has__**a**left-adjoint ( the Freyd adjoint functor theorem ).__Every__unit of length

__has__

**a**corresponding unit of area, namely the area of

**a**square with the given side length

**.**

__Every__ATM cell

__has__an 8-or 12-bit Virtual Path Identifier ( VPI ) and 16-bit Virtual Channel Identifier ( VCI ) pair defined in its header

**.**

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