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monogenous

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monogenous
monogenous, a. (məˈnɒdʒɪnəs) [f. Gr. µόνο-ς mono- + γέν-ος kind, origin, γεν- to grow, produce (see -gen) + -ous.] 1. Bot. (See quot. 1856.)1856 Mayne Expos. Lex., Monogenus, Bot., applied by Lessing to monocotyledonous plants, because they have but one surface of increase, which is central: monogen... Oxford English Dictionary
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Monogenous
Monogenous in mathematics may refer to: A synonym for cyclic in monogenous group, a synonym for cyclic group monogenous module, a synonym for cyclic wikipedia.org
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Monogenic
mathematics, a semigroup generated by a set containing only a single element Monogenic system, in classical mechanics, a physical system See also Monogenous wikipedia.org
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unigenous
† uˈnigenous, a. Geol. Obs.—1 [f. uni- + Gr. γέν-ος kind, origin: see -ous.] Of uniform structure. (Cf. monogenous a. 4.)1799 Kirwan Geol. Ess. 214 The unigenous limestone mountains of Carniola. Oxford English Dictionary
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Monogenetic
a cluster of volcanoes that only erupted once Monogenetic theory of pidgins in linguistics, a theory about the origin of creole languages See also Monogenous wikipedia.org
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monogeneous
monogeneous, a. (mɒnəʊˈdʒiːniːəs) [f. mod.L. monogene-us, f. Gr. µονογεν-ής (f. µόνο-ς mono- + γέν-ος kind, race) + -ous. Cf. F. monogéné.] 1. Of one race or family.1856 in Mayne Expos. Lex. 2. Biol. Generated in the same form as that of the parents.1890 Century Dict. s.v. 1891 Ibid. s.v. Trematoda,... Oxford English Dictionary
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polygenous
polygenous, a. (pəˈlɪdʒɪnəs) [irreg. f. Gr. πολυγεν-ής of many kinds (f. πολυ-, poly- + γένος kind) + -ous: perh. associated with words from L. such as indigenous, calcigenous, etc.] 1. Composed of constituents of different kinds; spec. in Geol. composed of various kinds of rocks.1799 Kirwan Geol. E... Oxford English Dictionary
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monogeny
monogeny (məˈnɒdʒɪnɪ) [f. Gr. µόνο-ς mono- + -geny. Cf. F. monogénie.] 1. The (theoretical) origination of mankind from one common pair of ancestors; also loosely, the theory of such origination, monogenism.1865 Anthrop. Rev. Oct. 362 This paper was entitled, ‘On Monogeny and Polygeny’. 1893 S. Lain... Oxford English Dictionary
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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 Cn, that is generated by a single monogenous group, avoiding the term "infinite cyclic group". wikipedia.org
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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. wikipedia.org
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divalent
divalent, a. Chem. (ˈdaɪvələnt, ˈdɪv-) [f. di-2 twice + L. valēnt-em, pr. pple. of valēre to be worth.] Combining with two atoms of hydrogen or other univalent element or radical; having two combining equivalents; also bivalent. A diatomic element, e.g. oxygen, is divalent; so is the highly complex ... Oxford English Dictionary
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Does $f\otimes \operatorname{Id} = \operatorname{Id}$ imply $f= \operatorname{Id}$? > Let $R$ be a commutative ring, and $X$ an $R$-module. If an $R$-endomorphism of $X$ satisfies $f\otimes \operatorname{Id}_X = \oper...
Your guess is right, it's indeed not true in general: Any choice of an $R$-module $X$ such that $X\otimes_R X=0$ and $f\neq\text{id}: X\to X$ gives a counterexample, for example you could take $R := {\mathbb Z}$, $X := {\mathbb Q}/{\mathbb Z}$ and $f = 2\cdot \text{id}$.
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