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Flat Cover
In algebra, a flat cover of a module ''M'' over a ring is a surjective homomorphism from a flat module ''F'' to ''M'' that is in some sense minimal. Any module over a ring has a flat cover that is unique up to (non-unique) isomorphism. Flat covers are in some sense dual to injective hulls, and are related to projective covers and torsion-free cover In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule is reduced to its ...s. Definitions The homomorphism ''F''→''M'' is defined to be a flat cover of ''M'' if it is surjective, ''F'' is flat, every homomorphism from flat module to ''M'' factors through ''F'', and any map from ''F'' to ''F'' commuting with the map to ''M'' is an automorphism of ''F''. History While projective covers for modules do not always exist, it was speculated that for general rings, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Module
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper '' Géometrie Algébrique et Géométrie Analytique''. See also flat morphism. Definition A module over a ring is ''flat'' if the following condition is satisfied: for every injective linear map \varphi: K \to L of -modules, the map :\varphi \otimes_R M: K \otimes_R M \to L \otimes_R M is also injective, where \varphi \otimes_R M is the map induced by k \otimes m \mapsto \varphi(k) \otimes m. For this definition, it is enough to restrict the injections \varphi to the inclusions of finitely generated ideals into . Equivalently, an -module is flat if the tensor product with is an exact functor; that is if, for every short exact sequence of - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Hull
In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in . Definition A module ''E'' is called the injective hull of a module ''M'', if ''E'' is an essential extension of ''M'', and ''E'' is injective. Here, the base ring is a ring with unity, though possibly non-commutative. Examples * An injective module is its own injective hull. * The injective hull of an integral domain is its field of fractions . * The injective hull of a cyclic ''p''-group (as Z-module) is a Prüfer group . * The injective hull of ''R''/rad(''R'') is Hom''k''(''R'',''k''), where ''R'' is a finite-dimensional ''k''-algebra with Jacobson radical rad(''R'') . * A simple module is necessarily the socle of its injective hull. * The injective hull of the residue field of a discrete valuation ring (R,\mathfrak,k) where \mathfrak = x\cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Cover
In the branch of abstract mathematics called category theory, a projective cover of an object ''X'' is in a sense the best approximation of ''X'' by a projective object ''P''. Projective covers are the dual of injective envelopes. Definition Let \mathcal be a category and ''X'' an object in \mathcal. A projective cover is a pair (''P'',''p''), with ''P'' a projective object in \mathcal and ''p'' a superfluous epimorphism in Hom(''P'', ''X''). If ''R'' is a ring, then in the category of ''R''-modules, a superfluous epimorphism is then an epimorphism p : P \to X such that the kernel of ''p'' is a superfluous submodule of ''P''. Properties Projective covers and their superfluous epimorphisms, when they exist, are unique up to isomorphism. The isomorphism need not be unique, however, since the projective property is not a full fledged universal property. The main effect of ''p'' having a superfluous kernel is the following: if ''N'' is any proper submodule of ''P'', then p(N) \n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion-free Cover
In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule is reduced to its zero element. In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero-divisors then the only module satisfying this condition is the zero module. Examples of torsion-free modules Over a commutative ring ''R'' with total quotient ring ''K'', a module ''M'' is torsion-free if and only if Tor1(''K''/''R'',''M'') vanishes. Therefore flat modules, and in particular free and projective modules, are to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
