Monogenous (other)
Monogenous in mathematics may refer to: * A synonym for cyclic in ** ''monogenous group'', a synonym for cyclic group ** ''monogenous module'', a synonym for cyclic module See also * Monogenic (other) * Monogenetic (other) Monogenetic may refer to: * Monogenetic in biology, of or pertaining to monogenesis (Mendelian inheritance) * Monogenetic volcanic field in geology, a cluster of volcanoes that only erupted once * Monogenetic theory of pidgins in linguistics, a t ...

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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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Cyclic (mathematics)
There are many terms in mathematics that begin with cyclic: * Cyclic chain rule, for derivatives, used in thermodynamics * Cyclic code, linear codes closed under cyclic permutations * Cyclic convolution, a method of combining periodic functions * Cycle decomposition (graph theory) * Cycle decomposition (group theory) * Cyclic extension, a field extension with cyclic Galois group * Graph theory: * Cyclic function, a periodic function **Cycle graph, a connected, 2-regular graph **Cycle graph (algebra), a diagram representing the cycles determined by taking powers of group elements **Circulant graph, a graph with cyclic symmetry **Cycle (graph theory), a nontrivial path in some graph from a node to itself **Cyclic graph, a graph containing at least one graph cycle * Cyclic group, a group generated by a single element * Cyclic homology, an approximation of K-theory used in non-commutative differential geometry * Cyclic module, a module generated by a single element * Cyclic notation, a ...
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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 binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a ''generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ...
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Cyclic Module
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 ...
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Monogenic (other)
Monogenic may refer to: * Monogenic signal, in the theory of analytic signals * Monogenic disorder, disease, inheritance, or trait, a single gene disorder resulting from a single mutated gene ** Monogenic diabetes, or maturity-onset diabetes of the young (MODY), forms of diabetes caused by mutations in an autosomal dominant gene * Monogenic field, in mathematics, an algebraic number field ''K'' * Monogenic function, a function in an algebra over a field * Monogenic polynomial, an alternate name for monic polynomial * Monogenic semigroup, in mathematics, a semigroup generated by a set containing only a single element * Monogenic system, in classical mechanics, a physical system See also * Monogenous (other) * Monogenetic (other) Monogenetic may refer to: * Monogenetic in biology, of or pertaining to monogenesis (Mendelian inheritance) * Monogenetic volcanic field in geology, a cluster of volcanoes that only erupted once * Monogenetic theory of pidgins in ling ...
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