Page "Algebraically compact module" ¶ 14
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## Some Related Sentences

Every and indecomposable
Every simple module is indecomposable, but the converse is in general not true.
Every injective module can be decomposed as direct sum of indecomposable injective modules.
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 simple module is indecomposable.

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 vector space is algebraically compact ( since it is pure-injective ).
Every module of the form H * is algebraically compact.
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 compact metric space is separable.
* 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.
* Sequentially compact: Every sequence has a convergent subsequence.
* Countably compact: Every countable open cover has a finite subcover.
* Limit point compact: Every infinite subset has an accumulation point.
Every compact metric space is complete, though complete spaces need not be compact.
Every entire function can be represented as a power series that converges uniformly on compact sets.
* Every compact metric space ( or metrizable space ) is separable.
* Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.
Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.
Every continuous function on a compact set is uniformly continuous.
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 ).
*( BCT2 ) Every locally compact Hausdorff space is a Baire space.
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.
*( BCT2 ) Every locally compact Hausdorff space is a Baire space.

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 simple module is cyclic, that is it is generated by one element.
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 module over a field or skew field is projective ( even free ).
* Every projective module is flat.
Every module possesses a projective resolution.
* Every module M has an injective hull.
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 module can focus on what it is designed for.
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 soldier in the army has, somewhere, relatives who are close to starvation.
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 calculation has been made independently by two workers and checked by one of the editors.
Every retiring person has a different situation facing him.
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 woman in the block has tried that ''.
: Every set has a choice function.
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 surjective function has a right inverse.
** 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.
** Antichain principle: Every partially ordered set has a maximal antichain.
** Every vector space has a basis.
* Every small category has a skeleton.
* 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 field has an algebraic closure.
** Every field extension has a transcendence basis.
** Every Tychonoff space has a Stone – Čech compactification.
* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =
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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