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2017 | OriginalPaper | Chapter

10. Extensions of Commutative Rings

Author : Alexey L. Gorodentsev

Published in: Algebra II

Publisher: Springer International Publishing

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Abstract

Everywhere in this section, the term “ring” means by default a commutative ring with unit. All ring homomorphisms are assumed to map the unit to the unit.

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previous chapter Categories and Functors

next chapter Affine Algebraic Geometry

See Sect. 9.6.1 of Algebra I, especially formula (9.29).

See Sect. 3.4.2 of Algebra I.

That is, lying in \(\mathrm{Mat}_{d}(\mathbb{Z}) \subset \mathrm{ Mat}_{d}(\mathbb{Q})\). Indeed, this is the original definition of algebraic integers, introduced in the nineteenth century by Dedekind.

See Sect. 4.1.2 of Algebra I.

See Sect. 5.4 of Algebra I.

That is, the monic polynomial μ _b ∈ Q _A[x] of minimal positive degree such that μ _b(b) = 0; see Sect. 8.1.3 of Algebra I.

Recall that the eigenvalues of an operator are among the roots of every polynomial annihilating the operator; see Exercise 15.13 of Algebra I.

See Lemma 5.3 on p. 103.

See Sect. 5.4.2 on p. 111.

See Sect. 8.1.3 of Algebra I.

See Proposition 10.3 on p. 229.

See Sect. 5.2.4 of Algebra I.

Generators of an algebra should be not confused with generators of a module. If elements e ₁, e ₂, …, e _m span a ring B over a subring A ⊂ B as a module, this means that B consists of finite A-linear combinations of these elements e _i, whereas if b ₁, b ₂, …, b _m span B as an A-algebra, then B is formed by finite linear combinations of various monomials \(b_{1}^{s_{1}}b_{2}^{s_{2}}\cdots b_{m}^{s_{m}}\).

If b is not algebraic, then \(\mathbb{k}[b] \simeq \mathbb{k}[x]\) is not a field.

See Sect. 4.1.2 of Algebra I.

Compare with the exchange lemma, Lemma 6.2, from Algebra I.

See Lemma 5.3 of Algebra I.

Recall that the content of a polynomial with coefficients in a unique factorization domain is the greatest common divisor of all the coefficients; see Sect. 5.4.4 of Algebra I.

Compare with Problem 14.1 from Algebra I.

See Sect. 5.1.2 of Algebra I.

[DK]

Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.MathSciNetCrossRef

[Fu]

Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.MATH

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Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.MATH

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Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.

Title: Extensions of Commutative Rings
Author: Alexey L. Gorodentsev
Publisher: Springer International Publishing
Book: Algebra II
Print ISBN: 978-3-319-50852-8

Electronic ISBN: 978-3-319-50853-5

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-50853-5_10

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