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Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module.
The completion of a finitely generated module M over a Noetherian ring R can be obtained by extension of scalars: M ^ = M ⊗ R R ^ . {\displaystyle {\widehat {M}}=M\otimes _{R}{\widehat {R}}.} Together with the previous property, this implies that the functor of completion on finitely generated R -modules is exact : it preserves short exact ...
For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...
Given an ideal I in a commutative ring R and an R-module M, the direct sum = / + is a graded module over the associated graded ring / +. A morphism f : N → M {\displaystyle f:N\to M} of graded modules, called a graded morphism or graded homomorphism , is a homomorphism of the underlying modules that respects grading; i.e., f ( N i ) ⊆ M ...
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.
The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits (which implies it right-splits, and corresponds to a direct product).
Existence: There exists an integer denoted a −1 such that aa −1 ≡ 1 (mod m) if and only if a is coprime with m. This integer a −1 is called a modular multiplicative inverse of a modulo m. If a ≡ b (mod m) and a −1 exists, then a −1 ≡ b −1 (mod m) (compatibility with multiplicative inverse, and, if a = b, uniqueness modulo 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.