Cyclic module

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In mathematics, more specifically in ring theory, a cyclic module is a module over a ring which is generated by one element. The concept is analogous to cyclic group, that is, a group which is generated by one element.

Definition[edit]

A left R-module M is called cyclic if M can be generated by a single element i.e. M = (x) = R x = {rx | rR} for some x in M. Similarly, a right R-module N is cyclic, if N = y R for some yN.

Examples[edit]

  • Every cyclic group is a cyclic Z-module.
  • Every simple R-module M is a cyclic module since the submodule generated by any nonzero element x of M is necessarily the whole module M.
  • 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 F[x]/(xλ)n; there may also be other cyclic submodules with different annihilators; see below.)

Properties[edit]

  • Given a cyclic R-module M which is generated by x then there exists a canonical isomorphism between M and R / AnnRx, where AnnRx denotes the annihilator of x in R.

See also[edit]

References[edit]