 Page "Noetherian ring" ¶ 69
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Every and injective Every embedding is injective and continuous. 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 module M has an injective hull. Every object in a Grothendieck category has an injective hull.

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 indecomposable, but the converse is in general not true. 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 of the form H * is algebraically compact. Every indecomposable algebraically compact module has a local endomorphism ring. 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 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 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. ** Well-ordering theorem: Every set can be well-ordered. 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 regular language is context-free because it can be described by a context-free grammar. 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 real number has a ( possibly infinite ) decimal representation ; i. e., it can be written as 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 positive integer n > 1 can be represented in exactly one way as a product of prime powers: 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 morpheme can be classified as either free or bound. 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 document window is an object with which the user can work. 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 ordered field can be embedded into the surreal numbers. * 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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