Ads
related to: mathews triax draw length modules
Search results
Results From The WOW.Com Content Network
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.
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.
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).
A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5 ...
The ring R is left hereditary if and only if all left modules have projective resolutions of length at most 1. This is equivalent to saying that the left global dimension is at most 1. Hence the usual derived functors such as E x t R i {\displaystyle \mathrm {Ext} _{R}^{i}} and T o r i R {\displaystyle \mathrm {Tor} _{i}^{R}} are trivial for i ...
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.
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).
Indeed, if M is a module of finite length, then, by induction on length, it has a finite indecomposable decomposition = =, which is a decomposition with local endomorphism rings. Now, suppose we are given an indecomposable decomposition M = ⨁ i = 1 m N i {\textstyle M=\bigoplus _{i=1}^{m}N_{i}} .