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The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded -algebra.
An Artinian module is a module that satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps. Graded A graded module is a module with a decomposition as a direct sum M = ⨁ x M x over a graded ring R = ⨁ x R x such that R x M y ⊆ M x+y for all x and y ...
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, [1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of ...
For a finite projective module over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma. [3] For the general case, the proof (both the original as well as later one) consists of the following two steps: Observe that a projective module over an arbitrary ring is a direct sum of countably generated projective modules.
A module P is projective if and only if every short exact sequence of modules of the form . is a split exact sequence.That is, for every surjective module homomorphism f : B ↠ P there exists a section map, that is, a module homomorphism h : P → B such that f h = id P .
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
Semisimple decomposition: a direct sum of simple modules. Indecomposable decomposition: a direct sum of indecomposable modules. A decomposition with local endomorphism rings [5] (cf. #Azumaya's theorem): a direct sum of modules whose endomorphism rings are local rings (a ring is local if for each element x, either x or 1 − x is a unit).
M/tM is a finitely generated torsion free module, and such a module over a commutative PID is a free module of finite rank, so it is isomorphic to: for a positive integer n. Since every free module is projective module, then exists right inverse of the projection map (it suffices to lift each of the generators of M/tM into M).
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