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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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Are simple commutative monoids monogeneous? Let $M$ be a simple commutative monoid. Is there a surjective monoid morphism $\mathbb N\to M$? If non-monogeneous simple commutative monoids do exist, what's known about ...

Let $L=\\{0,1\\}$, considered as a commutative monoid under multiplication. If $M$ is any commutative monoid, there is a homomorphism $f:M\to L$ which sends all invertible elements to $1$ and all non-invertible elements to $0$. If $M$ is simple, then either $f$ must be an isomorphism or $f$ must fai...

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monogeneity

monogeneity Math. (mɒnədʒɪˈniːɪtɪ) [Formed as next + -ity.] The state or condition of being monogeneous.1906 Athenæum 20 Jan. 83/3 The following papers were communicated [on Jan. 11]: ‘On the Monogeneity of an Algebraic Function’, by Dr. H. F. Baker, [etc.]. Oxford English Dictionary

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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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Name of some specific orders in number fields Let $\mathbb{K}/\mathbb{Q}$ a number field. For an integer $\theta\in\mathfrak{o}_\mathbb{K}$, one can define an order $$ \mathbb{Z}[\theta] = \oplus_i \mathbb{Z} \theta^...

The computer algebra system `magma` calls such a thing an _equation order_. It is isomorphic to the ring $\mathbb{Z}[x]/(f(x))$, where $f$ is the minimal polynomial for $\theta$ over $\mathbb{Q}$. The specific basis $\\{ \theta^i \\}$ would be called a _power basis_.

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is $\mathbb{Q}$ finitely generated as a $\mathbb{Z}_{(p)}$-algebra? I know that $\mathbb{Q}$ is not finitely generated as a $\mathbb{Z}$-algebra (and thus also not finitely generated as a $\mathbb{Z}$-module) how abou...

It is true — it is even a monogeneous algebra: $$\mathbf Q=\mathbf Z_{(p)}[X]/(pX-1).$$

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monogeneous

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monogeneous

Are simple commutative monoids monogeneous? Let $M$ be a simple commutative monoid. Is there a surjective monoid morphism $\mathbb N\to M$? If non-monogeneous simple commutative monoids do exist, what's known about ...

monogeneity

monogenous

Name of some specific orders in number fields Let $\mathbb{K}/\mathbb{Q}$ a number field. For an integer $\theta\in\mathfrak{o}_\mathbb{K}$, one can define an order $$ \mathbb{Z}[\theta] = \oplus_i \mathbb{Z} \theta^...

is $\mathbb{Q}$ finitely generated as a $\mathbb{Z}_{(p)}$-algebra? I know that $\mathbb{Q}$ is not finitely generated as a $\mathbb{Z}$-algebra (and thus also not finitely generated as a $\mathbb{Z}$-module) how abou...