In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept...

In abstract algebra, a cyclic group or monogenous group is a group, denoted C n {\displaystyle C_{n}} (also frequently Z n {\displaystyle \mathbb {Z} _{n}}...

(d_{2})\supseteq \cdots \supseteq (d_{n})} such that M is isomorphic to the sum of cyclic modules: M ≅ ⨁ i R / ( d i ) = R / ( d 1 ) ⊕ R / ( d 2 ) ⊕ ⋯ ⊕ R / ( d n ) ...

coefficients from the ring R. Cyclic A module is called a cyclic module if it is generated by one element. Free A free R-module is a module that has a basis, or...

the module M is called a Noetherian module. If a module is generated by one element, it is called a cyclic module. Let R be an integral domain with K...

and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length...

geometry Cyclic module, a module generated by a single element Cyclic notation, a way of writing permutations Cyclic number, a number such that cyclic permutations...

algebra, cyclic decomposition refers to writing a finitely generated module over a principal ideal domain as the direct sum of cyclic modules and one free...

forbidden to exclude trivial cyclic subspaces). The resulting list of polynomials are called the invariant factors of (the K[X]-module defined by) the matrix...

abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...

finitely generated R-module, then M {\displaystyle M} is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic...

is a faithful Noetherian module over A then A is Noetherian as well. Over a commutative ring, every cyclic Artinian module is also Noetherian, but over...

at most countable. cyclic A module is called a cyclic module if it is generated by one element. D A D-module is a module over a ring of differential operators...

we make V a k[t]-module. The structure theorem then says V is a direct sum of cyclic modules, each of which is isomorphic to the module of the form k [...

equivalently a module R / I , {\displaystyle R/I,} is a cyclic representation of R (cyclic meaning generated by 1 element as an R-module; this generalizes...

Warfield: it states that every finitely presented module over a serial ring is a direct sum of cyclic uniserial submodules (and hence is serial). If additionally...

refer to: A synonym for cyclic in monogenous group, a synonym for cyclic group monogenous module, a synonym for cyclic module Monogenic (disambiguation)...

cyclic module. Similarly over a semiperfect ring, every indecomposable projective module is a PIM, and every finitely generated projective module is...

groups are exactly the cyclic groups of prime order. The concepts of abelian group and Z {\displaystyle \mathbb {Z} } -module agree. More specifically...

the Brown–Gitler spectrum is a spectrum whose cohomology is a certain cyclic module over the Steenrod algebra. Brown–Gitler spectra are defined by the isomorphism:...

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...

finitely-generated R-module is a direct sum of these. Note that this is simple if and only if n = 1 (or p = 0); for example, the cyclic group of order 4,...

well-defined interfaces. Often modules form a directed acyclic graph (DAG); in this case a cyclic dependency between modules is seen as indicating that these...

to modules. Given an R-module A and a collection of elements a1, ..., an of A, the submodule of A spanned by a1, ..., an is the sum of cyclic modules R...

representation can fail to be irreducible without splitting as a direct sum. For a cyclic group C generated by g of order n, the matrix form of an element of K[C]...

_{N})} , is given by a Mahler measure (or is infinite). In the case of a cyclic module M = R / ⟨ F ⟩ {\displaystyle M=R/\langle F\rangle } for a non-zero polynomial...

her characterization of semisimple rings in terms of properties of cyclic modules. Osofsky also established a logical equivalence between the continuum...

direct product of minimal normal subgroups. As an example, consider the cyclic group Z12 with generator u, which has two minimal normal subgroups, one...

such as where (1) L is a flat K-module, (2) L is torsion-free as an abelian group, (3) L is a direct sum of cyclic modules (or all its localizations at prime...

groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory. If G is a finite cyclic group acting...