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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
Cyclic_module
Generalization of vector spaces from fields to rings
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
Module_(mathematics)
Mathematical group that can be generated as the set of powers of a single element
In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently Z {\displaystyle \mathbb {Z} } n or Zn, not to be confused
Cyclic_group
Statement in abstract algebra
(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 )
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Index of articles associated with the same name
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
Cyclic_(mathematics)
In algebra, module with a finite generating set
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
Finitely_generated_module
Type of module over a ring
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
Simple_module
Canonical form of matrices over a field
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
Frobenius_normal_form
cardinality is 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
Glossary_of_module_theory
Algebraic structure with addition and multiplication
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 [
Ring_(mathematics)
Algebraic structure
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
Principal_ideal_domain
Mathematical object in abstract algebra
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
Injective_module
Problematic interdependence of software modules
software engineering, a circular dependency (or cyclic dependency) is a relation between two or more modules which either directly or indirectly depend on
Circular_dependency
Topics referred to by the same term
refer to: A synonym for cyclic in monogenous group, a synonym for cyclic group monogenous module, a synonym for cyclic module Monogenic (disambiguation)
Monogenous
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:
Brown–Gitler_spectrum
Topics referred to by the same term
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
Cycle_decomposition
Commutative group (mathematics)
groups are exactly the cyclic groups of prime order. The concepts of abelian group and Z {\displaystyle \mathbb {Z} } -module agree. More specifically
Abelian_group
Representation theory of groups
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]
Regular_representation
cyclic module. Similarly over a semiperfect ring, every indecomposable projective module is a PIM, and every finitely generated projective module is
Principal indecomposable module
Principal_indecomposable_module
Organizing code into modules
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
Modular_programming
Multi particle state space
Fock space. In representation theory, Fock spaces can be described as cyclic modules generated by a distinguished vector, the vacuum. Annihilation operators
Fock_space
Set of a ring's prime ideals
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
Spectrum_of_a_ring
Module which satisfies the descending chain condition on submodules
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
Artinian_module
In linear algebra, generated subspace
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
Linear_span
H_{*}(C)} is finitely generated; in particular, only finitely many cyclic modules of the form Z / p s {\displaystyle \mathbb {Z} /p^{s}} can appear as
Bockstein_spectral_sequence
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,
Indecomposable_module
Measure of polynomial height
_{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
Mahler_measure
Decomposition of an algebraic structure
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
Composition_series
Index of articles associated with the same name
direct product of minimal normal subgroups. As an example, consider the cyclic group Z12 with generator u, which has two minimal normal subgroups, one
Socle_(mathematics)
American mathematician
her characterization of semisimple rings in terms of properties of cyclic modules. Osofsky also established a logical equivalence between the continuum
Barbara_L._Osofsky
Software programming object-oriented design methodology
modules to low-level, dependency modules are reversed, thus rendering high-level modules independent of the low-level module implementation details. The principle
Dependency inversion principle
Dependency_inversion_principle
Explicitly describes the universal enveloping algebra of a Lie algebra
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
Poincaré–Birkhoff–Witt theorem
Poincaré–Birkhoff–Witt_theorem
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
Serial_module
Secondary metabolides
can synthesize only one type of peptide. Nonribosomal peptides often have cyclic and/or branched structures, can contain non-proteinogenic amino acids including
Nonribosomal_peptide
Chemical compound
leading to increased afferent vagus nerve stimulation. Cereulide is a cyclic dodecadepsipeptide resembling valinomycin; it contains three repeats of
Cereulide
Studies linear representations of finite groups over fields of positive characteristic
defect group is when the latter is cyclic. Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure
Modular_representation_theory
Branch of mathematics
points by other structures, such as representations, modules, traces, states, K-theory classes, cyclic cocycles or categories. This shift is one reason that
Noncommutative_geometry
Operation that pairs a left and a right R-module into an abelian group
of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction
Tensor_product_of_modules
Notion in abstract algebra
Example 3.35). The injective hull of a cyclic p {\displaystyle p} -group (as Z {\displaystyle \mathbb {Z} } -module) is a Prüfer group (Lam 1999, Example
Injective_hull
Operation in algebra
three ways to change the coefficient ring of a module; namely, for a right R-module M and a right S-module N, one can form f ∗ M = M ⊗ R S {\displaystyle
Change_of_rings
Commutative group in which all nonzero elements have the same order
non-negative integer (sometimes called the group's rank). Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript
Elementary_abelian_group
Mexican mathematician (1933-2014)
; Gitler, Samuel (1973). "A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra". Topology. 12 (3): 283–295. doi:10
Samuel_Gitler_Hammer
In algebra, integer associated to a module
theory. The zero module is the only one with length 0. Modules with length 1 are precisely the simple modules. The length of the cyclic group Z / n Z {\displaystyle
Length_of_a_module
American mathematician (1926–2021)
; Gitler, Samuel (1973). "A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra". Topology. 12 (3): 283–295. doi:10
Edgar_H._Brown
American mathematician (1917–1955)
I.S.; Kaplansky, Irving (1951). "Rings for which every module is a direct sum of cyclic modules". Mathematische Zeitschrift. 54 (1): 97–101. doi:10.1007/BF01179851
Irvin_Cohen
Branch of mathematics that studies algebraic structures
theory) Simple module, Semisimple module Indecomposable module Artinian module, Noetherian module Homological types: Projective module Projective cover
List of abstract algebra topics
List_of_abstract_algebra_topics
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
Herbrand_quotient
Computational problem used in cryptography
x_{n-1})} Micciancio introduced cyclic lattices in his work in generalizing the compact knapsack problem to arbitrary rings. A cyclic lattice is a lattice that
Short integer solution problem
Short_integer_solution_problem
Measure of the structural complexity of a software program
command. Cyclomatic complexity may also be applied to individual functions, modules, methods, or classes within a program. One testing strategy, called basis
Cyclomatic_complexity
Number in {..., –2, –1, 0, 1, 2, ...}
\mathbb {Z} } under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to Z {\displaystyle \mathbb
Integer
Subgraph induced by all nodes linked to a given node of a graph
(k)-(ultra)-homogeneous graph is locally (k)-(ultra)-homogeneous. A graph is locally cyclic if every neighbourhood is a cycle. For instance, the octahedron is the unique
Neighbourhood_(graph_theory)
Mathematical term in group theory
isomorphic to a finite cyclic p-group or to a Prüfer group. The Prüfer p-group is the unique infinite p-group that is locally cyclic (every finite set of
Prüfer_group
Knot invariant
his polynomial. Let K be a knot in the 3-sphere. Let X be the infinite cyclic cover of the knot complement of K. This covering can be obtained by cutting
Alexander_polynomial
Terpene hydrocarbon
(/ˈlɪmənˌiːn/) is a colorless liquid aliphatic hydrocarbon classified as a cyclic monoterpene, and is the major component in the fragrance and essential oil
Limonene
German assault rifle
NATO Action Gas-operated short-stroke piston, rotating bolt Rate of fire Cyclic rate: 700–900 rounds/min Muzzle velocity 882 m/s (2,890 ft/s) Effective firing range
Heckler_&_Koch_G95
Computer network protocol
as with CC-C. In contrast to CC-A and CC-B, the complete communication (cyclic and acyclic) between controller and device takes place on Ethernet layer
Profinet
Monster and modular connection
known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky
Monstrous_moonshine
Elements taken to zero by a homomorphism
considered a particular kind of module kernel when the underlying ring is the integers. Let G {\displaystyle G} be the cyclic group on 6 elements { 0 , 1
Kernel_(algebra)
local class field theory: The module A is the group of non-zero complex numbers, and G is either trivial or is the cyclic group of order 2 generated by
Class_formation
Mathematical construct
is denoted by b p ( ξ ) {\displaystyle b_{p}(\xi )} . The number of cyclic modules in the torsion part is denoted by q p ( ξ ) {\displaystyle q_{p}(\xi
Novikov_ring
Multinational wheeled armoured fighting vehicle
drive module and interchangeable mission modules which allow several configurations to meet different operational requirements. The drive module has been
Boxer (armoured fighting vehicle)
Boxer_(armoured_fighting_vehicle)
case of the integers and abelian groups a pure projective module amounts to a direct sum of cyclic groups. Divisible group Fuchs, L (1970), Infinite Abelian
Pure_subgroup
Wireless communication technology
chirp spread spectrum (CSS) modulation. Each symbol is represented by a cyclic shifted chirp over the bandwidth centered around the base frequency. The
LoRa
Wireless communications protocol
positioning applications. NearLink employs the Cyclic Prefix-Orthogonal Frequency Division Multiplexing (Cyclic Prefix-OFDM) waveform to address latency issues
NearLink
Conversion process for computer data
functions can preserve sharing and handle cyclic data, which can be configured by a flag. Several Perl modules available from CPAN provide serialization
Serialization
Set with associative invertible operation
multiplicative group of a field is necessarily cyclic. See Lang 2002, Theorem IV.1.9. The notions of torsion of a module and simple algebras are other instances
Group_(mathematics)
Phenomenon in crystallization
is often caused by accidents during growth. In the tetragonal system, cyclical contact twins are the most commonly observed type of twin, such as in rutile
Crystal_twinning
American prototype 5.56 mm rotary-barreled machine gun
telescope. The Six-Pak consisted of the XM214, the power module, and the ammunition module consisted of two 500-round, factory packed, and disposable
XM214_Microgun
V is a subgroup of the multiplicative group of the field K, and hence a cyclic group. It follows that G consists of roots of unity of order dividing n
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
Square matrix constructed from a monic polynomial
A:F^{n}\to F^{n}} makes F n {\displaystyle F^{n}} a cyclic F [ A ] {\displaystyle F[A]} -module, having a basis of the form { v , A v , … , A n − 1 v
Companion_matrix
Topics referred to by the same term
cohomology of a bimodule over a Banach algebra Cyclic homology of an associative algebra Group cohomology of a module over a group ring or a representation of
Cohomology_of_algebras
Algebraic structure associated with a topological space
to be modules over a coefficient ring R {\displaystyle R} and taking the boundary maps d n {\displaystyle d_{n}} to be R {\displaystyle R} -module homomorphisms
Homology_(mathematics)
Tools for studying groups based on techniques from algebraic topology
actions of a group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements
Group_cohomology
Hypothetical scenario for the ultimate fate of the universe
bang. This could potentially repeat forever in a phenomenon known as a cyclic universe. Richard Bentley, a churchman and scholar, sent a letter to Isaac
Big_Crunch
Chemical compound
II (GrsB), to give a product as a cyclic decapeptide. Within the biosynthetic pathway, there are total of five modules that specifically recognize, activate
Gramicidin_S
Relation between relative derivatives of three variables
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule, Euler's chain rule, or the reciprocity theorem, is a
Triple_product_rule
Set of finitely supported functions from a group to a ring
then the module structure of the group ring R G {\displaystyle RG} is in fact a vector space over R {\displaystyle R} . 1. Let G = C3, the cyclic group of
Group_ring
Mathematical group based upon a finite number of elements
structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral
Finite_group
Operation that combines groups
that disjoint union plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless
Free_product
Group type in algebra
single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z. A locally cyclic group is a group in which
Finitely_generated_group
Any liquid polymerized siloxane with organic side chains
external coolant loop and radiators of the International Space Station Zvezda module, which rejects heat in the vacuum of space. The class of silicone oils known
Silicone_oil
Area of mathematics
general be irreducible. The modules so constructed are called Specht modules, and every irreducible does arise inside some such module. There are now fewer irreducibles
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Type of group in abstract algebra
multiplication. Cyclic groups are those that are generated by a single permutation. When a permutation is represented in cycle notation, the order of the cyclic subgroup
Symmetric_group
Spore-forming organisms
using traveling waves of cyclic AMP. There is an amplification of cyclic AMP when they aggregate. Pre-stalk cells move toward cyclic AMP, but pre-spore cells
Slime_mold
Result due to Kummer on cyclic extensions of fields that leads to Kummer theory
abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer
Hilbert's_Theorem_90
Short chains of 2–50 amino acids
many different modules to perform a diverse set of chemical manipulations on the developing product. These peptides are often cyclic and can have highly
Peptide
question of the existence of a normal integral basis is part of Galois module theory. Let F ⊂ K {\displaystyle F\subset K} be a Galois extension with
Normal_basis
Mathematical theorem
ring and let U be a simple right R-module. If u is a non-zero element of U, u • R = U (where u • R is the cyclic submodule of U generated by u). Therefore
Jacobson_density_theorem
Enzyme with key regulatory roles in most cells
diphosphate-lyase (cyclizing; 3′,5′-cyclic-AMP-forming). It catalyzes the following reaction: ATP ⟶ {\displaystyle \longrightarrow } 3′,5′-cyclic AMP + diphosphate It
Adenylyl_cyclase
Cryptography secured against quantum computers
(NIST) finalized its first post-quantum cryptography standards, including module-lattice-based key encapsulation and digital signature schemes, providing
Post-quantum_cryptography
Concept in ring theory and homological algebra
projective dimensions of all cyclic right A-modules; the supremum of the set of projective dimensions of all finite right A-modules; the supremum of the injective
Global_dimension
Standardized way to automatically access information about a memory module
is a standardized way to automatically access information about a memory module. Earlier 72-pin SIMMs included five pins that provided five bits of parallel
Serial_presence_detect
Graph with oriented edges
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed
Directed_graph
Result about when a matrix can be diagonalized
\lambda \mapsto \lambda } . A vector φ {\displaystyle \varphi } is called a cyclic vector for A {\displaystyle A} if the vectors φ , A φ , A 2 φ , … {\displaystyle
Spectral_theorem
Equivalence relation on rings
preserved include being free, and being cyclic. Many ring-theoretic properties are stated in terms of their modules, and so these properties are preserved
Morita_equivalence
Group of unitary matrices
fundamental group of U ( n ) {\displaystyle \operatorname {U} (n)} is infinite cyclic for all n {\displaystyle n} : π 1 ( U ( n ) ) ≅ Z . {\displaystyle \pi
Unitary_group
Algebra of formal sums
may equivalently be called free Z {\displaystyle \mathbb {Z} } -modules, the free modules over the integers. Lattice theory studies free abelian subgroups
Free_abelian_group
American assault rifle
Rifle M27 IAR with ACOG Squad Day Optic and AN/PEQ-16A weapon-mounted laser module with visible laser, infrared targeting laser, infrared illuminator, and
M27_Infantry_Automatic_Rifle
German series of submachine guns/personal defence weapons
hardened steel penetrator instead of softer copper or lead. The MP7 has a cyclic rate of fire of around 950 rounds per minute (which is around 15.8 rounds
Heckler_&_Koch_MP7
Mathematical structure
{\#} kG} is also denoted by A # G {\displaystyle A\mathop {\#} G} . The cyclic homology of Hopf smash products has been computed. However, there the smash
Group_Hopf_algebra
CYCLIC MODULE
CYCLIC MODULE
Boy/Male
Anglo, British, English
With Royal Might
Boy/Male
Tamil
Janardhana | ஜநாரà¯à®¤à®¾à®¨à®¾
Lord Krishna, One who helps people, Liberator from the cycle of birth and death
Janardhana | ஜநாரà¯à®¤à®¾à®¨à®¾
Boy/Male
Hindu
Free from the cycle of births and deaths
Boy/Male
Tamil
Jaramarana Varjita | ஜராமாஂரநா வரà¯à®œà¯€à®¤à®¾
Free from the cycle of births and deaths
Jaramarana Varjita | ஜராமாஂரநா வரà¯à®œà¯€à®¤à®¾
Boy/Male
Tamil
Janardana | ஜநாரà¯à®¤à®¨
Lord Krishna, One who helps people, Liberator from the cycle of birth and death
Janardana | ஜநாரà¯à®¤à®¨
Boy/Male
Hindu
Lord Krishna, One who helps people, Liberator from the cycle of birth and death
Male
Irish
Irish name CAILTE means "the thin man." This is the name of a character from the Fenian cycle.
Boy/Male
Hindu
Lord Krishna, One who helps people, Liberator from the cycle of birth and death
Girl/Female
Hindu, Indian, Traditional
The Periphery or Rim of a Wheel or Cycle
Boy/Male
Tamil
Janardan | ஜநாரà¯à®¤à®¨
Lord Krishna, One who helps people, Liberator from the cycle of birth and death
Janardan | ஜநாரà¯à®¤à®¨
Boy/Male
Assamese, Hindu, Indian, Marathi
The Healer; Vishnu; Who Cures the Disease of Birth and Death Cycles
Surname or Lastname
English
English : nickname from Middle English loller ‘indolent fellow’, a derivative of lolle ‘to droop, dangle, or loll’.English : nickname from Middle English lollere ‘mumbler’, bestowed on a pious person or on a Lollard (a follower of the 14th-century religious reformer John Wyclif).
Surname or Lastname
English
English : habitational name from a place in Cheshire named Kelsall, from the Middle English personal name Kell + Old English halh ‘nook or corner of land’, or possibly from Kelshall in Hertfordshire, which is named with an Old English personal name Cylli + Old English hyll ‘hill’, or even Kelsale in Suffolk, named with an Old English personal name Cēl(i) or Cēol + Old English halh.
Girl/Female
American, Arabic, Australian, British, Chinese, English
Stone of the Colic; The Gemstone Jade; Green in Colour
Boy/Male
English
royal.
Boy/Male
Hindu
Lord Krishna, One who helps people, Liberator from the cycle of birth and death
Boy/Male
Hindu
Lord Krishna, One who helps people, Liberator from the cycle of birth and death
Boy/Male
Tamil
Janardhan | ஜநாரà¯à®¤à®¨
Lord Krishna, One who helps people, Liberator from the cycle of birth and death
Janardhan | ஜநாரà¯à®¤à®¨
Boy/Male
Hindu, Indian, Marathi
Vishnu; The Healer; Who Cures the Disease of Birth and Death Cycles
Male
Spanish
Spanish name of Germanic origin, possibly GUIOMAR means "famous in battle." In the 13th century Vulgate Cycle of Arthurian romance, Sir Guiomar is the proud and beautiful knight of the crystal stream.
CYCLIC MODULE
CYCLIC MODULE
Male
Hindi/Indian
Variant spelling of Hindi Govind, GOBIND means "cow-finder."
Boy/Male
Indian
Shadows
Girl/Female
Hindu, Indian, Sanskrit, Telugu
Name of Rani Laxmibai; Earring Containing a Jewel
Biblical
a perfume; sacrifice of labor
Boy/Male
Indian
Pezan mard means holly Man
Girl/Female
Arabic
Originated from Madina
Surname or Lastname
English
English : patronymic from Fowle.
Boy/Male
German, Teutonic
From Windsor
Girl/Female
Arabic, Australian
Falconess
Boy/Male
British, English, Irish
Long Lived
CYCLIC MODULE
CYCLIC MODULE
CYCLIC MODULE
CYCLIC MODULE
CYCLIC MODULE
n.
The act or practice of using a cycle; cycling.
imp. & p. p.
of Cycle
n.
A cycler.
a.
Pertaining to the Dog Star; as, the cynic, or Sothic, year; cynic cycle.
n.
One who rides a bicycle or tricycle; a cycler, or cyclist.
a.
Adhering to a fixed circle of legends; cyclic; hence, mean; inferior. See Cyclic poets, under Cyclic.
a.
Containing cysts; cystose; as, cystic sarcoma.
n.
The act, art, or practice, of riding a cycle, esp. a bicycle or tricycle.
a.
Of or pertaining to a cycle or circle; moving in cycles; as, cyclical time.
a.
Of or pertaining to matter; material; corporeal; as, hylic influences.
n.
One entire round in a circle or a spire; as, a cycle or set of leaves.
p. pr. & vb. n.
of Cycle
n.
A mean or inferior poet, perhaps from his habit of wandering around as a stroller; an itinerant poet. Also, a name given to the cyclic poets. See under Cyclic, a.
a.
Having the form of, or living in, a cyst; as, the cystic entozoa.
a.
Alt. of Cyclical
v. i.
To ride a bicycle, tricycle, or other form of cycle.
a.
Of or pertaining to colic; affecting the bowels.
a.
Of or pertaining to the colon; as, the colic arteries.
a.
See Cystic.
v. i.
To pass through a cycle of changes; to recur in cycles.