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finite length A module of finite length is a module that admits a (finite) composition series. finite presentation 1. A finite free presentation of a module M is an exact sequence where are finitely generated free modules. 2.
In algebra, the length of a module over a ring is a generalization of the dimension of a vector space which measures its size. [1] page 153 It is defined to be the length of the longest chain of submodules. For vector spaces (modules over a field), the length equals the dimension.
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).
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces.The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules.
A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules. For a module M, the following are equivalent: M is semisimple; i.e., a direct sum of irreducible modules. M is the sum of its irreducible submodules.
In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times the restriction of s for any f in O(U) and s in F(U).
Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups.These are the cyclic groups of prime order.. If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal.
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution [1]) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object of this category.