Algebra Notes [lecture notes]

45 Pages·2015·0.434 MB·English

by  Ralph Freese, William DeMeo

Algebra Notes
Ralph Freese and William DeMeo
March 10, 2015
Contents
I Fall 2010: Universal Algebra & Group Theory 4
1 Universal Algebra 5
1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Subalgebras and Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Congruence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Quotient Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Direct Products of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
II Rings, Modules and Linear Algebra 12
2 Rings 13
2.1 The ring M (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
n
2.2 Factorization in Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Rings of Frations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Euclidean Domain and the Eucidean Algorithm . . . . . . . . . . . . . . . . . . . . . 17
2.5 Polynomial Rings, Gauss’ Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Irreducibility Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Modules 21
3.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Finitely Generated Modules over a PID . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.1 Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Projective, Injective and Flat Modules; Exact Sequences . . . . . . . . . . . . . . . . 37
1
CONTENTS CONTENTS
III Fields 41
4 Basics 42
A Prerequisites 43
A.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2
CONTENTS CONTENTS
Primary Textbook: Jacobson, Basic Algebra [4].
Supplementary Textbooks: Hungerford, Algebra [3]; Dummitt and Foote. Abstract Algebra [1];
Primary Subject: Classical algebra systems: groups, rings, fields, modules (including vector
spaces). Also a little universal algebra and lattice theory.
List of Notation
• AA, the set of maps from a set A into itself.
• Aut(A), the group of automorphisms of an algebra A.
• End(A), the set of endomorphisms an algebra A.
• Hom(A,B), the set of homomorphism from an algebra A into an algebra B.
• Con(A), the set of congruence relations of an algebra A.
• ConA, the lattice of congruence relations of an algebra A.
• Eq(A), the set of equivalence relations of a set A.
• EqA, the lattice of equivalence relations of a set A.
• Sub(A), the set of subalgebras of an algebra A.
• SubA, the lattice of subalgebras of an algebra A.
• Sg (X), the subuniverse generated by a set X ⊆ A.
A
• N = {1,2,...}, the set of natural numbers.
• Z = {...,−1,0,1,...}, the ring of integers.
• R = (−∞,∞), the real number field.
• C, the complex number field.
• Q, the rational number field.
3
Part I
Fall 2010: Universal Algebra & Group Theory
4
1 UNIVERSAL ALGEBRA
1 Universal Algebra
1.1 Basic concepts
A (universal) algebra is a pair
A = (cid:104)A;F(cid:105) (1.1)
where A is a nonempty set and F = {f : i ∈ I} is a set of finitary operations on A; that is,
i
f : An → A for some n ∈ N. A common shorthand notation for (1.1) is (cid:104)A;f (cid:105) . The number n
i i i∈I
is called the arity of the operation f .
i
Thus, the arity of an operation is the number of operands upon which it acts, and we say that
f ∈ F isann-aryoperationonAiff mapsAn intoA. Anoperationiscallednullary (orconstant)
if its arity is zero. Unary, binary, and ternary operations have arities 1, 2, and 3, respectively.
Example 1.1. If A = R and f : R×R → R is the map f(a,b) = a+b, then (cid:104)A;f(cid:105) is an algebra
with a single binary operation. Many more examples will be given below.
An algebra A is called unary if all of its operations are unary. An algebra A is finite if |A|
is finite and trivial if |A| = 1. Given two algebras A and B, we say that B is a reduct of A if
both algebras have the same universe and A (resp. B) can be obtained from B (resp. A) by adding
(resp. removing) operations.
A better approach: An operation symbol f is an object that has an associated arity, which we’ll
denote arity(f). A set of operation symbols F is called a similarity type. An algebra of similarity
type F is a pair A = (cid:104)A;FA(cid:105), where FA = {fA : f ∈ F} and fA is an operation on A of arity
arity(f).
Example 1.2. Consider the set of integers Z with operation sysbols F = {+,·,−,0,1}, which have
Z Z
respectivearities{2,2,1,0,0}. Theoperation+ istheusualbinaryaddition, while− isnegation:
Z Z
a (cid:55)→ −a. The constants 0 and 1 are nullary operations. Of course we usually just write + for
Z
+ , etc.
1.2 Subalgebras and Homomorphisms
Suppose A = (cid:104)A;FA(cid:105) is an algebra. We call the nonempty set A the universe of A. If a subset
B ⊆ A is closed under all operations in FA, we call B a subuniverse of A. By closed under all
operations we mean the following: for each f ∈ FA (say f is n-ary), we have f(b ,...,b ) ∈ B,
0 n−1
for all b ,...,b ∈ B.
0 n−1
If B (cid:54)= ∅ is a subuniverse of (cid:104)A;FA(cid:105), and if we let1 FB = {f (cid:22) B : f ∈ FA}, then the algebra
B = (cid:104)B;FB(cid:105) is called a subalgebra of A. If B is a subalgebra of A, we denote this fact by B (cid:54) A.
Similarly, we write B (cid:54) A if B is a subuniverse of A. We denote the set of all subalgebras of A by
Sub(A). Note that universe of an algebra is not allowed to be empty. However, the empty set can
be a subuniverse (if A has no nullary operations)..
Theorem 1.3. If A (cid:54) A, i ∈ I, then (cid:84)A is a subuniverse.
i i
1Here f (cid:22)B denotes restriction of the function f to the set B (see Appendix Sec. A.2).
5
1.3 Direct Products 1 UNIVERSAL ALGEBRA
If S is a nonempty subset of A, the subuniverse generated by S, denoted Sg (S) or (cid:104)S(cid:105) is
A
the smallest subuniverse of A containing the set S. When (cid:104)S(cid:105) = A, we say that S generates A.
Theorem 1.4. If S ⊆ A, then Sg (S) = (cid:104)S(cid:105) = (cid:84){B (cid:54) A : S ⊆ B}.
A
Define {S } recursively as follows:
i
S = S;
0
S = {f(a ,...,a ) : f is a k-ary basic operation of A and a ∈ S }.
i+1 1 k i i
Theorem 1.5. Sg (S) = (cid:104)S(cid:105) = (cid:83)∞ S .
A i=0 i
Let A = (cid:104)A;FA(cid:105) and B = (cid:104)B;FB(cid:105) be algebras of the same type F, and let F denote the set
n
of n-ary operation symbols in F. Consider a mapping ϕ : A → B and operation symbol f ∈ F ,
n
and suppose that for all a ,...a ∈ A the following equation holds:
0 n−1
ϕ(fA(a ,...,a )) = fB(ϕ(a ),...,ϕ(a )).
0 n−1 0 n−1
Then ϕ is said to respect the interpretation of f. If ϕ respects the interpretation of every f ∈ F,
then we call ϕ a homomorphism from A into B, and we write ϕ ∈ Hom(A,B), or simply,
ϕ : A → B.
1.3 Direct Products
The direct product of two sets A and A , denoted A ×A , is the set of all ordered pairs2 (cid:104)a ,a (cid:105)
0 1 0 1 0 1
such that a ∈ A and a ∈ A . That is, we define
0 0 1 1
A ×A := {(cid:104)a ,a (cid:105) : a ∈ A ,a ∈ A }.
0 1 0 1 0 0 1 1
More generally, A ×···×A is the set of all sequences of length n with ith element in A . That
0 n−1 i
is,
A ×···×A := {(cid:104)a ,...,a (cid:105) : a ∈ A ,...,a ∈ A }.
0 n−1 0 n−1 0 0 n−1 n−1
Equivalently, A ×···×A it is the set of all functions with domain {0,1,...,n−1} and range
0 n−1
n−1
(cid:83)
A . More generally still, let {A : i ∈ I} be an indexed family of sets. Then the direct product
i i
i=1
of the A is
i
(cid:89) (cid:91)
A := {f | f : I → A with f(i) ∈ A }.
i i i
i∈I i∈I
When A = A = ··· = A, we often use the shorthand notation A2 := A×A and An := A×···×A
0 1
(n terms).
Question: How do you know (cid:81) A (cid:54)= ∅, even supposing I (cid:54)= ∅ and A (cid:54)= ∅ for all i ∈ I.3
i i
i∈I
2For the definition of ordered pair, consult the appendix.
3Answer: Each f “chooses” an element from each A , but when the A are all different and I is infinite, we may
i i
not be able to do this. The Axiom of Choice (AC) says you can.
Go¨delprovedthattheACisconsistentwiththeotheraxiomsofsettheory. Cohenprovedthatthenegationofthe
AC is also consistent.
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1.4 Relations 1 UNIVERSAL ALGEBRA
1.4 Relations
A k-ary relation R on a set A is a subset of the cartesian product Ak.
Example 1.6.
(a) A = R (the line) and R = {a ∈ R2 : a is rational}.
(b) A = R2 (the plane) and R = {(a,b,c) ∈ R2 ×R2 ×R2 : a, b, c lie on a line}; i.e. tripples of
points which are colinear.
(c) A = R2 and R = {(a,b) ∈ R2 ×R2 : a = (a ,a ), b = (b ,b ), a2 +a2 = b2 +b2}. This is an
1 2 1 2 1 2 1 2
equivalence relation. The equivalence classes are circles centered at (0,0).
(d) A = R2 and R = “(cid:54) on each component” = {(a,b) ∈ R2×R2 : a (cid:54) b , a (cid:54) b }.
1 1 2 2
The relation in the last example above is a partial order; that is, (cid:54) satisfies, for all a,b,c,
1. a (cid:54) a (reflexive)
2. a (cid:54) b, b (cid:54) a ⇒ a = b (anti-symmetric)
3. a (cid:54) b, b (cid:54) c ⇒ a (cid:54) c (transitive)
A relation R on a set A is an equivalence relation if it satisfies, for all a,b,c,
1. aRa (reflexive)
2. aRb ⇒ bRa (symmetric)
3. aRb, bRc ⇒ aRc (transitive)
We denote the set of all equivalence relations on a set A by Eq(A).
A partition of a set A is a collection Π = {A : i ∈ I} of non-empty subsets of A such that
i
(cid:91)
A = A and A ∩A = ∅ for all pairs i (cid:54)= j in I.
i i j
i∈I
Facts and notation:
1. The A are called “blocks.”
i
2. Each partition Π determines an equivalence relation – namely, the relation θ defined by aθb
if and only if a and b are in the same block of Π.
3. Conversely, if θ is an equivalence relation on A (θ ∈ Eq(A)), we denote the equivalence class
of θ containing a by a/θ := {b ∈ A : aθb} and the set A/θ := {a/θ : a ∈ A} of all θ classes is
a partition of A.
4. “an SDR” (add something here)
7
1.5 Congruence Relations 1 UNIVERSAL ALGEBRA
1.5 Congruence Relations
Let A and B be sets and let ϕ : A → B be any mapping. We say that a pair (cid:104)a ,a (cid:105) ∈ A2 belongs
0 1
to the kernel of ϕ, and we write (cid:104)a ,a (cid:105) ∈ kerϕ, just in case ϕ(a ) = ϕ(a ). Thus, to every map
0 1 0 1
ϕ there corresponds a relation ∼ = kerϕ, defined by
ϕ
a ∼ a(cid:48) ⇔ ϕ(a) = ϕ(a(cid:48)).
ϕ
We leave it as an exercise to prove
Proposition 1.7. kerϕ is an equivalence relation.
If ∼ is an equivalence relation on a set A, then a/∼ denotes the equivalence class containing
a; that is, a/∼ := {a(cid:48) ∈ A : a(cid:48) ∼ a}. The set of all equivalence classes of ∼ in A is denoted A/∼.
That is, A/∼ = {a/∼ : a ∈ A}.
Let A = (cid:104)A;FA(cid:105) and B = (cid:104)B;FB(cid:105) be algebras of the same type and let ϕ : A → B be
any mapping. As above, ϕ determines the equivalence relation ∼ = kerϕ on A. A congruence
ϕ
relation on A is an equivalence relation ∼ arising from a homomorphism ϕ : A → B, for some
ϕ
B. We denote the set of all congruence relations on A by ConA.
To reiterate, θ ∈ ConA iff θ = kerϕ for some homomorphism ϕ of A. It is easy to check that
this is equivalent to: θ ∈ ConA iff θ ∈ Eq(A) and
(cid:104)a ,a(cid:48)(cid:105) ∈ θ (0 (cid:54) i < n) ⇒ (cid:104)f(a ,...,a ),f(a(cid:48),...,a(cid:48) )(cid:105) ∈ θ, (1.2)
i i 0 n−1 0 n−1
for all f ∈ F and all a ,...,a ,a(cid:48),...,a(cid:48) ∈ A. Equivalently,4
n 0 n−1 0 n−1
ConA = Eq(A)∩Sub(A×A).
We record this fact as
Theorem 1.8. Suppose A = (cid:104)A;F(cid:105) is an algebra and θ is an equivalence relation on the set A.
Then θ is congruence relation on the algebra A if and only if for every n-ary basic operation f ∈ F
and for all a ,...,a ,a(cid:48),...,a(cid:48) ∈ A,
0 n−1 0 n−1
a θa(cid:48) (0 (cid:54) i < n) ⇒ f(a ,...,a )θf(a(cid:48),...,a(cid:48) ).
i i 0 n−1 0 n−1
In fact, we can replace the final implication in the theorem with: for each 0 (cid:54) i < n,
a θa(cid:48) ⇒ f(a ,...,a ,...a )θf(a ,...,a(cid:48),...,a ).
i i 0 i n−1 0 i n−1
Proof sketch: Recall that a/θ denotes the equivalence class of θ which contains a, and A/θ denotes
the full set of θ classes: A/θ = {a/θ : a ∈ A}. For every operation fA on A (say it is n-ary), we
define
fA/θ(a /θ,...,a /θ) := fA(a ,...,a )/θ.
1 n 1 n
One easily checks that this is well-defined and that the map a (cid:55)→ a/θ is a homomorphism of A onto
the algebra A/θ := (cid:104)A/θ,FA/θ(cid:105), where FA/θ := {fA/θ : fA ∈ F}. Finally, note that the kernel of
this homomorphism is θ.
4The direct product algebra A×A is defined below in section 1.7.
8
1.6 Quotient Algebras 1 UNIVERSAL ALGEBRA
1.6 Quotient Algebras
LetA = (cid:104)A;F(cid:105)beanalgebra. Recall,F denotesthesetofoperationsymbols,andtoeachoperation
symbol f ∈ F there corresponds an arity, and the set of arities determines the similarity type of
the algebra. (We might do better to denote the algebra by (cid:104)A;FA(cid:105), since the algebra is not defined
until we have associated to each operation symbol f ∈ F an actual operation fA on A, but this is
a technical point, and we will often denote two algebras of the same similarity type as (cid:104)A;F(cid:105) and
(cid:104)B;F(cid:105) with the understanding that the meaning of F depends on the context.)
Let θ ∈ ConA be a congruence relation. The quotient algebra A/θ is an algebra with the
same similarity type as A, with universe A/θ = {a/θ : a ∈ A}, and operation symbols F, where for
each (k-ary) symbol f ∈ F the operation fA/θ is defined as follows: for (a /θ,...,a /θ) ∈ (A/θ)k,
1 k
fA/θ(a /θ,...,a /θ) = fA(a ,...,a )/θ.
1 k 1 k
(As mentioned above, F denotes the set of operation symbols of the similarity type of the algebra,
and it is “overloaded” in the sense that we write A = (cid:104)A;F(cid:105) and A/θ = (cid:104)A/θ,F(cid:105), and for each
f ∈ F the corresponding operation in these algebras is interpreted appropriately – i.e., as fA or
fA/θ.)
1.7 Direct Products of Algebras
Abovewedefineddirectproductsofsets. Wenowdefinedirectproductsofalgebras. LetA = (cid:104)A;F(cid:105)
and B = (cid:104)B;F(cid:105) be two algebras of the same similarity type. The direct product A×B is an
algebra of the same type as A and B, with universe A×B = {(a,b) : a ∈ A,b ∈ B}, and operation
symbols F. To each (k-ary) symbol f ∈ F corresponds an operation fA×B defined as follows: for
((a ,b ),...,(a ,b )) ∈ (A×B)k,
1 1 k k
fA×B((a ,b ),...,(a ,b )) = (fA(a ,...,a ),fB(b ,...,b )). (1.3)
1 1 k k 1 k 1 k
(cid:81)
This definition can be easily extended to the direct product A of any collection of algebras
i
{A : i ∈ I}, and we leave it to the reader to write down the defining property of the operations,
i
which is completely analogous to (1.3).
∼ (cid:81)
When all the algebras are the same – that is, when A = A, for some A – we call A the
i i
direct power of A. When the set I is finite, say, I = 1,2,...,n, we have alternative notations for
the direct power, namely,
(cid:89)
A = A ×A ×···×A = An.
i 1 2 n
The constructions of this subsection section and the preceeding one are often combined to give
direct products of quotient algebras. For instance, if θ and θ are two congruences of A, the
1 2
algebra A/θ ×A/θ has the same similarity type as A, and its universe is
1 2
A/θ ×A/θ = {(a/θ ,b/θ ) : a,b ∈ A}.
1 2 1 2
The operation symbols are again F, and to each (k-ary) symbol f ∈ F corresponds an operation
fA/θ1×A/θ2 defined as follows: for ((a1/θ1,b1/θ2),...,(ak/θ1,bk/θ2)) ∈ (A/θ1×A/θ2)k,
fA/θ1×A/θ2((a /θ ,b /θ ),...,(a /θ ,b /θ )) = (fA/θ1(a /θ ,...,a /θ ),fA/θ2(b /θ ,...,b /θ ))
1 1 1 2 k 1 k 2 1 1 k 1 1 2 k 2
= (fA(a ,...,a )/θ ,fA(b ,...,b )/θ ).
1 k 1 1 k 2
9
1.8 Lattices 1 UNIVERSAL ALGEBRA
1.8 Lattices
A relational structure is a set A and a collection of (finitary) relations on A. A partially
ordered set, or poset, is a set A together with a partial order (Sec. 1.4) (cid:54) on it, denoted (cid:104)A,(cid:54)(cid:105).
Let (cid:104)A,(cid:54)(cid:105) be a poset and let B be a subset of the set A. An element a in A is an upper bound
for B if b (cid:54) a for every b in B. An element a in A is the least upper bound of B, denoted (cid:87)B,
or supremum of B (supB), if a is an upper bound of B, and b (cid:54) c for every b in B implies a (cid:54) c
(i.e., a is the smallest among the upper bounds of B). Similarly, a is a lower bound of B provided
a (cid:54) b for all b in B, and a is the greatest lower bound of B ((cid:86)B), or infimum of B (infB) if a
is a lower bound and is above every other lower bound of B.
Let a,c be two elements in the poset A. We say c covers a, or a is covered by c provided a (cid:54) c
and whenever a (cid:54) b (cid:54) c it follows that a = b or b = c. We use the notation a ≺ c to denote that c
covers a.
A lattice is a partially ordered set (cid:104)L;(cid:54)(cid:105) such that for each pair a,b ∈ L there is a least upper
bound, denoted a∨b := lub{a,b}, and a greatest lower bound, denoted a∧b := glb{a,b}, contained
in L. A lattice can also be viewd as an algebra (cid:104)L;∨,∧(cid:105) where ∨, called “join,” and ∧, “meet,” are
binary operations satisfying
1. x∨x = x and x∧x = x (idempotent)
2. x∨y = y∨x and x∧y = y∧x (commutative)
3. x∨(y∨z) = (x∨y)∨z (associative)
4. x∨(y∧x) = x and x∧(y∨x) = x (absorbtive)
Posetsingeneral,andlatticesinparticular,canbevisualizedusingaso-calledHasse diagram.
The Hasse diagram of a poset (cid:104)A,(cid:54)(cid:105) is a graph in which each element of the set A is denoted by
a vertex, or “node” of the graph. If a ≺ b then we draw the node for b above the node for
a, and join them with a line segment. The resulting diagram gives a visual description of the
relation (cid:54), since a (cid:54) b holds iff for some finite sequence of elements c ,...,c in A we have
1 n
a = c ≺ c ≺ ··· ≺ c = b. Some examples appear in the figures below.
1 2 n
Figure 1: Hasse diagrams
a b
c d
Note that the first two examples in Figure 1 depict the same poset, which illustrates that
Hasse diagrams are not uniquely determined. Also, note that the poset represented in the first two
diagrams is a lattice. In contrast, the poset depicted in the third diagram is not a lattice since
10