Cyclic module explained
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 is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element.
Definition
A left R-module M is called cyclic if M can be generated by a single element i.e. for some x in M. Similarly, a right R-module N is cyclic if for some .
Examples
- 2Z as a Z-module is a cyclic module.
- In fact, every cyclic group is a cyclic Z-module.
- Every simple R-module M is a cyclic module since the submodule generated by any non-zero element x of M is necessarily the whole module M. In general, a module is simple if and only if it is nonzero and is generated by each of its nonzero elements.
- If the ring R is considered as a left module over itself, then its cyclic submodules are exactly its left principal ideals as a ring. The same holds for R as a right R-module, mutatis mutandis.
- If R is F[''x''], the ring of polynomials over a field F, and V is an R-module which is also a finite-dimensional vector space over F, then the Jordan blocks of x acting on V are cyclic submodules. (The Jordan blocks are all isomorphic to ; there may also be other cyclic submodules with different annihilators; see below.)
Properties
- Given a cyclic R-module M that is generated by x, there exists a canonical isomorphism between M and, where denotes the annihilator of x in R.
See also
References
. B. Hartley . Brian Hartley . T.O. Hawkes . Rings, modules and linear algebra . limited . Chapman and Hall . 1970 . 0-412-09810-5 . 77, 152.
