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LECTURE NOTES for MA20217: ALGEBRA 2B Contents 1. Rings 2 LECTURE NOTES FOR MA20217: ALGEBRA 2B ALASTAIR CRAW (2013/14) Abstract. This course introduces abstract ring theory and provides a thorough struc- ture theory of linear operators on finite dimensional vector spaces. Contents 1. Rings 2 1.1. A reminder on groups 2 1.2. Definitions and basic properties of rings 3 1.3. Examples of rings 5 1.4. When do equivalence classes form a ring? 7 1.5. Subrings and ideals 9 2. Ring homomorphisms 12 2.1. Definitions and examples 12 2.2. The fundamental isomorphism theorem 15 2.3. The characteristic of a ring with 1 17 2.4. The Chinese remainder theorem 18 3. Factorisation in integral domains 21 3.1. Integral domains and Euclidean domains 21 3.2. Principal ideal domains 22 3.3. Irreducible elements in an integral domain 23 3.4. Unique factorisation domains 25 3.5. General polynomial rings 27 3.6. Field of fractions and Gauss' lemma 28 4. Associative algebras with 1 over a field 30 4.1. Algebras 30 4.2. Constructing field extensions 32 4.3. Normed R-algebras 35 4.4. Application to number theory 36 5. The structure of linear operators 39 5.1. Minimal polynomials 39 5.2. Invariant subspaces 41 5.3. Primary Decomposition 42 5.4. The Jordan Decomposition over C 45 5.5. Jordan normal form over C 47 1 1. Rings 1.1. A reminder on groups. Informally, a ring is simply a set equipped with `sensible' notions of addition and multiplication that are compatible. We would like the definition to be broad enough to include examples like the set of n×n matrices over a fixed field with the usual matrix addition and multiplication, the set of polynomials with coefficients in some fixed field with the usual polynomial addition and multiplication, and the integers. At the same time we want the definition to be somewhat restricted so that we can build a general theory that deals with all these examples at once. Before introducing the formal definition of a ring (and recalling that of a group), recall that a binary operation on a set S is a function f : S × S ! S: The binary operations that crop up here are typically addition, denoted +, or multipli- cation, denoted ·. We write a + b rather than +(a; b), and a · b rather than ·(a; b). Definition 1.1 (Group). A group is a pair (G; ∗), where G is a set, ∗ is a binary operation on G and the following axioms hold: (a) (The associative law) (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a; b; c 2 G: (b) (Existence of an identity) There exist an element e 2 G with the property that e ∗ a = a and a ∗ e = a for all a 2 G: (c) (The existence of an inverse) For each a 2 G there exists b 2 G such that a ∗ b = b ∗ a = e: If it is clear from the context what the group operation ∗ is, one often simply refers to the group G rather than to the pair (G; ∗). Remarks 1.2. Both the identity element and the inverse of a given element are unique: (1) if e; f 2 G are two elements satisfying the identity property from (b) above, then f = e ∗ f = e; where the first identity follows from the fact that e satisfies the property and the latter from the fact that f satisfies the property. (2) Given a 2 G, if b; c 2 G are both elements satisfying (c) above, then b = b ∗ e = b ∗ (a ∗ c) = (b ∗ a) ∗ c = e ∗ c = c: This unique element b is called the inverse of a. It is often denoted a−1. Definition 1.3 (Abelian group). A group (G; ∗) is abelian if a∗b = b∗a for all a; b 2 G. 2 The binary operation in an abelian group is often written as +, in which case the identity element is denoted 0, and the inverse of an element a 2 G is denoted −a 2 G. Definition 1.4 (Subgroup). A nonempty subset H of a group G is called a subgroup of G iff (1.1) 8 a; b 2 H; we have a ∗ b−1 2 H: This version of the definition is great when you want to show that a subset is a subgroup, because there's so little to check. Despite this, we have (see Algebra 1A, Prop 6.3): Lemma 1.5. A nonempty subset H of a group (G; ∗) is a subgroup if and only if (H; ∗) is a group. Proof. Let H be a subgroup of (G; ∗). Since H is nonempty, there exists a 2 H and hence e = a ∗ a−1 2 H by equation (1.1). For a 2 H, apply condition (1.1) to the elements e; a 2 H to see that a−1 = e∗a−1 2 H. Also, for a; b 2 H, we've just shown that b−1 2 H, so applying condition (1.1) to the elements a; b−1 2 H gives a ∗ b = a ∗ (b−1)−1 2 H. In particular, ∗ is a binary operation on H, and since (G; ∗) is a group, the operation ∗ on H is associative. For the converse, let H be a subset of G such that (H; ∗) is a group. Then the identity element e 2 H, so H is nonempty. Let a; b 2 H. Then b−1 lies in H since H −1 is a group, and since ∗ is a binary operation on H we have a ∗ b 2 H as required. 1.2. Definitions and basic properties of rings. We now move on to rings. Definition 1.6 (Ring). A ring is a triple (R; +; ·), where R is a set with binary operations +: R × R ! R (a; b) 7! a + b and ·: R × R ! R (a; b) 7! a · b such that the following axioms hold: (1)( R; +) is an abelian group. Write 0 for the (unique) additive identity, and −a for the (unique) additive inverse of a 2 R, so (a + b) + c = a + (b + c) for all a; b; c 2 R; a + 0 = a for all a 2 R; a + b = b + a for all a; b 2 R; a + (−a) = 0 for all a 2 R: (2)( R; ·) satisfies the associative law, that is, we have (a · b) · c = a · (b · c) for all a; b; c 2 R; (3) R satisfies the distributive laws: a · (b + c) = (a · b) + (a · c) for all a; b; c 2 R; (b + c) · a = (b · a) + (c · a) for all a; b; c 2 R: 3 Notation 1.7. We often omit · and write ab instead of a · b. For simplicity we often avoid brackets when there is no ambiguity. Here the same conventions hold as for real numbers, i.e., that · has priority over +. For example ab + ac stands for (a · b) + (a · c) and not (a · (b + a)) · c. One also writes a2 for a · a and 2a for a + a and so on. Lemma 1.8. In any ring (R; +; ·), we have (1) a · 0 = 0 and 0 = 0 · a for all a 2 R; and (2) a · (−b) = −(a · b) and −(a · b) = (−a) · b for all a; b 2 R. Proof. For (1), let a 2 R. Since 0 is an additive identity, one of the distributive laws gives a · 0 = a · (0 + 0) = a · 0 + a · 0: Adding −(a · 0) on the left on both sides gives −(a · 0) + a · 0 = −(a · 0) + a · 0 + a · 0: The left hand side is zero, and the associativity law gives 0 = (−(a · 0) + a · 0) + a · 0 = 0 + a · 0 = a · 0 as required. The second identity is similar. To prove (2), note that a · b + a · (−b) = a · (b + (−b)) = a · 0 = 0: This means that a · (−b) is the additive inverse of ab, that is, a · (−b) = −(a · b). The second identity is similar. Definition 1.9 (Rings with additional properties). Let (R; +; ·) be a ring. Then: (1) R a ring with 1 if there is an element 1 := 1R 2 R satisfying a · 1 = 1 · a = a for all a 2 R: (2) R is a commutative ring if a · b = b · a for all a; b 2 R: (3) R a division ring if it is a ring with 1 such that for all a 2 R n f0g; there exists b 2 R such that ab = 1 = ba: (4) R is a field if it is a commutative division ring in which 0 6= 1. Remark 1.10. If R is a ring with 1, then 1 is the unique multiplicative identity. The same argument as before works, i.e., if 1¯ was another multiplicative identity, then 1¯ = 1¯ · 1 = 1. Definition 1.11 (Unit). Let R be a ring with 1. An element a 2 R is called a unit if it has a multiplicative inverse, i.e., if there exists b 2 R such that a · b = b · a = 1. Remarks 1.12. (1) In a division ring, every nonzero element is a unit. (2) The multiplicative inverse of a unit is unique, see Remark 1.2(2) for the argument. We denote the multiplicative inverse by a−1. 4 (3) If 0 is a unit then Lemma 1.8(1) implies that 1 = 0·0−1 = 0. Therefore, for a 2 R, we have a = a · 1 = a · 0 = 0, i.e., R is the zero ring f0g. Definition 1.13 (Group of units).
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