Abstract
In this chapter, K by default means an arbitrary commutative ring with unit and \(\mathbb{k}\) means an arbitrary field. A K-module always means a unital module over K.
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Notes
- 1.
See Definition 6.2 on p. 124.
- 2.
Also called a K- linear map.
- 3.
See Sect. 2.6 on p. 31.
- 4.
See Sect. 9.6 on p. 220.
- 5.
See Sect. 8.1.3 on p. 175.
- 6.
See Sect. 5.2.2 on p. 107.
- 7.
See Sect. 5.3 on p. 109.
- 8.
Note that d is linearly independent, because K has no zero divisors: λ d = 0 ⇒ λ = 0.
- 9.
Recall that the greatest common divisor of elements in a principal ideal domain is a generator of the ideal spanned by those elements. It is unique up to multiplication by invertible elements of K (see Sect. 5.3.2 on p. 110).
- 10.
Compare with Exercise 5.17 on p. 111.
- 11.
Compare with Sect. 8.4 on p. 182.
- 12.
See Definition 5.1 on p. 104.
- 13.
Note that the same power p m can appear several times in the collection of elementary divisors if it appears in the prime factorizations of several of the f i .
- 14.
See Problem 5.8 on p. 120.
- 15.
That is, such that (q) ≠ (p), or equivalently, \(\mathop{\mathsf{GCD}}\nolimits (q,p) = 1\) (see Exercise 5.17 on p. 111).
- 16.
Considered up to multiplication by invertible elements.
- 17.
Recall that in an additive abelian group, the order of an element w is the minimal \(n \in \mathbb{N}\) such that nw = 0. If there is no such n, we set \(\mathop{\mathrm{ord}}\nolimits (w) = \infty\) (see Sect. 3.6.1 on p. 62).
- 18.
Modules possessing these properties are called Noetherian (compare with Lemma 5.1 on p. 104).
- 19.
See Sect. 5.1.2 on p. 104.
- 20.
See 14.10 on p. 355.
- 21.
See Sect. 14.1.8 on p. 343.
- 22.
See Definition 14.6 on p. 355.
- 23.
Compare with Problem 12.11 on p. 304.
- 24.
That is, points with integer coordinates lying inside but not on the boundary of \(\Pi\), faces of \(\Pi\), and edges of Π respectively.
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Gorodentsev, A.L. (2016). Modules over a Principal Ideal Domain. In: Algebra I. Springer, Cham. https://doi.org/10.1007/978-3-319-45285-2_14
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