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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
See Sect. 3.4.2 of Algebra I.
- 3.
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.
- 4.
See Sect. 4.1.2 of Algebra I.
- 5.
See Sect. 5.4 of Algebra I.
- 6.
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.
- 7.
Recall that the eigenvalues of an operator are among the roots of every polynomial annihilating the operator; see Exercise 15.13 of Algebra I.
- 8.
See Lemma 5.3 on p. 103.
- 9.
See Sect. 5.4.2 on p. 111.
- 10.
See Sect. 8.1.3 of Algebra I.
- 11.
See Proposition 10.3 on p. 229.
- 12.
See Sect. 5.2.4 of Algebra I.
- 13.
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 \(b_{1}^{s_{1}}b_{2}^{s_{2}}\cdots b_{m}^{s_{m}}\).
- 14.
If b is not algebraic, then \(\mathbb{k}[b] \simeq \mathbb{k}[x]\) is not a field.
- 15.
See Sect. 4.1.2 of Algebra I.
- 16.
Compare with the exchange lemma, Lemma 6.2, from Algebra I.
- 17.
See Lemma 5.3 of Algebra I.
- 18.
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.
- 19.
Compare with Problem 14.1 from Algebra I.
- 20.
See Sect. 5.1.2 of Algebra I.
References
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Gorodentsev, A.L. (2017). Extensions of Commutative Rings. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-50853-5_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-50852-8
Online ISBN: 978-3-319-50853-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)