In
mathematics, more specifically in
ring theory, a cyclic module or monogenous module[1] 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. M = (x) = Rx = {rx | r ∈ R} for some x in M. Similarly, a right R-module N is cyclic if N = yR for some y ∈ N.
Every
simpleR-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.[2]
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.
Given a cyclic R-module M that is generated by x, there exists a canonical isomorphism between M and R / AnnRx, where AnnRx denotes the
annihilator of x in R.
In
mathematics, more specifically in
ring theory, a cyclic module or monogenous module[1] 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. M = (x) = Rx = {rx | r ∈ R} for some x in M. Similarly, a right R-module N is cyclic if N = yR for some y ∈ N.
Every
simpleR-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.[2]
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.
Given a cyclic R-module M that is generated by x, there exists a canonical isomorphism between M and R / AnnRx, where AnnRx denotes the
annihilator of x in R.