mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, more specifically in

ring theory In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...

, a cyclic module or monogenous module is a

module over a ring In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...

that is generated by one element. The concept is a generalization of the notion of a

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

, that is, an

Abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...

(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

is a cyclic Z-module. * Every

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

''R''-module ''M'' is a cyclic module since the

submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...

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 ideal In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it ...

s as a ring. The same holds for ''R'' as a right ''R''-module,

mutatis mutandis ''Mutatis mutandis'' is a Medieval Latin phrase meaning "with things changed that should be changed" or "once the necessary changes have been made". It remains unnaturalized in English and is therefore usually italicized in writing. It is used i ...

. * If ''R'' is ''F'' 'x'' the

ring of polynomials In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...

over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''F'', and ''V'' is an ''R''-module which is also a

finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

over ''F'', then the

Jordan block In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has t ...

s of ''x'' acting on ''V'' are cyclic submodules. (The Jordan blocks are all

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

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''. *Every module is a sum of cyclic submodules.

References

* * * {{Lang Algebra, edition=3, pages=147–149 Module theory

Definition

Examples

Properties

See also

References