non-commutative algebra 2015–2016

122 Pages·2016·0.6 MB·English

NON-COMMUTATIVE ALGEBRA
2015–2016
E. Jespers
Departement of Mathematics
Vrije Universiteit Brussel
ii
Contents
1 Groups and Rings 3
1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Ring constructions . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Group constructions . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Finite completely 0-simple semigroups . . . . . . . . . . . . . 11
1.5 Structure of finite semigroups. . . . . . . . . . . . . . . . . . . 18
1.6 Structure of linear semigroups. . . . . . . . . . . . . . . . . . 19
2 Semisimplicity Problems 21
2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Structural Results: revisited . . . . . . . . . . . . . . . . . . . 24
2.3 The Jacobson Radical . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Group algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Semigroup Algebras . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Crossed Products . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.7 Polynomial rings and graded rings . . . . . . . . . . . . . . . 45
3 Noetherian Characterization Problem 51
3.1 Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 The zero divisor problem 59
4.1 Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Prime and semiprime rings 81
5.1 Group algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Semigroup algebras . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Maximal Orders 89
7 Units of Group Rings 101
7.1 Constructions of units in ZG . . . . . . . . . . . . . . . . . . 101
7.2 Rational group algebras of finite cyclic groups . . . . . . . . . 104
7.3 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.4 Free subgroups in U(ZG) . . . . . . . . . . . . . . . . . . . . 111
iii
CONTENTS 1
Index 118
2 CONTENTS
Chapter 1
Groups and Rings
Thebasictheoryofgroupsandringshasbeendealtwithinpreviouscourses.
In the first section we recall some definitions.
1.1 Basic Concepts
Definition1.1.1. Asemigroupisasetwithanassociativebinaryoperation.
Sometimes this object is denoted (S,∗), or simply S, where S is the set and
∗ is the operation. An element θ of S is calles a zero element if sθ = θs = θ
for all s ∈ S.
Definition 1.1.2. A monoid is a semigroup S which has a unit element 1,
i.e. 1s = s1 = s for each s ∈ S. This object is sometimes denoted (S,∗,1).
Definition 1.1.3. A group G is a monoid in which every element is invert-
ible. An element g ∈ G is said to be invertible in G if there exists g(cid:48) ∈ G
such that gg(cid:48) = g(cid:48)g = 1. The element g(cid:48) is called the inverse of g and is
denoted g−1. The group G is said to be abelian, or commutative, if gh = hg
for all g,h ∈ G.
Mostly we will use the multiplicative notation for groups. However,
for abelian groups we will use the additive notation, with + for the group
operation, 0 for the unit element, and −g for the inverse of g.
An element g of a group G is torsion (or periodic) if gn = 1 for some
positive integer. The group is said to be torsion-free if the identity element
is the only torsion element.
Definition 1.1.4. A subgroup N of a group G is normal if gNg−1 = N for
all g ∈ G. The quotient group of G by N is denoted G/N.
Now we recall some definitions of sets with two operations.
Definition 1.1.5. A ring is a set R together with two operations, the ad-
dition + and the multiplication ., and two distinguished elements 0 and 1,
such that the following conditions are satisfied:
3
4 CHAPTER 1. GROUPS AND RINGS
1. (R,+,0) is an abelian group,
2. (R,.,1) is a monoid,
3. the distributive laws: a(b+c) = ab+ac and (b+c)a = ba+ca for all
a,b,c ∈ R.
If, moreover, ab = ba for all a,b ∈ R, then R is said to be commutative (or
abelian).
Sometimes we also will consider rings without a multiplicative identity.
The trivialringisthesingleton{0}. Inthisring1 = 0. Inallotherrings
the latter does not happen. A subring of a ring is a subset which is itself
a ring under the induced multiplication and addition and with the same
distinguished elements 1 and 0 (although sometimes one does not demand
that the subring has the same 1).
Adomainisaringinwhicheachproductofnonzeroelementsisnonzero.
A ring R is called a division ring, or skew field, if (R\{0},.,1) is a group. A
commutative domain is called an integral domain; a commutative division
ring is called a field.
A function f : R → S from a ring R to a ring S is said to be a ring
homomorphism if f is a homomorphism of the abelian groups (R,+) and
(S,+), and the monoids (R,.) and (S,.).
1.2 Ring constructions
We first give several natural ring constructions.
Example 1.2.1. 1. TheringofintegersZ, thefieldsofrationalnumbers
Q, real numbers R, complex numbers C.
2. Matrix Rings.
Let R be a ring, n a positive integer. The ring of n×n matrices over
the ring R, denoted M (R), is the complete set of of all n×n arrays
n
or matrices
 
r r ... r
11 12 1n
M =  r21 r22 ... r2n ,
 ... ... ... ... 
r r ... r
n1 n2 nn
orsimplydenotedasM = (r ). Additionandmultiplicationaregiven
ij
according to the rules:
(a )+(b ) = (a +b ) and
ij ij ij ij
n
(cid:88)
(a )(b ) = (c ) where c = a b .
ij ij ij ij ik kj
k=1
Note for n > 1, M (R) always is non-commutative.
n
1.2. RING CONSTRUCTIONS 5
3. Upper Triangular Matrices.
The ring of all matrices (r ) ∈ M (R) with r = 0 for i > j.
ij n ij
4. Polynomial Rings.
Let R be a ring. The polynomial ring (in one variable X) over R is
the set of all formal sums
f(X) = r +r X +...r Xn
0 1 n
where each r ∈ R. If r (cid:54)= 0, then n is called the degree of the
i n
polynomial f(X), denoted deg(f(X)), and r is called the leading
n
coefficient of f(X). The degree of the zero polynomial is by definition
−∞. Addition and multiplication are given according to the rules:
n n n
(cid:88) (cid:88) (cid:88)
a Xi+ b Xi = (a +b )Xi and
i i i i
i=1 i=1 i=1
n m n+m
(cid:88) (cid:88) (cid:88) (cid:88)
( a Xi)( b Xj) = ( (a b ))Xk
i j i j
i=1 j=1 k=1 i+j=k
(in the latter we agree that a = 0 if i > n, and b = 0 if j > m). More
i j
generally, the polynomial ring in the variables X ,...,X is defined
1 n
inductively by the rule
R[X ,...,X ] = (R[X ,...,X ])[X ].
1 n 1 n−1 n
It follows from the definition that every element (called a polynomial)
is of the form
(cid:88)
f = r Xi1...Xin,
i1...in 1 n
where the r ∈ R are uniquely determined as the coefficient of
i1...in
Xi1...Xin. Note that formally the above is an infinite sum, but in
1 n
factonlyfinitelymanyofthecoefficientsr arenonzero. Notealsothat
i
the X ’s commute among each other and also with the coefficients.
i
An element of the form mi = X1i1...Xnin is called a monomial, and its
total degree (or simply degree), denoted deg(m ), is the sum (cid:80)n i .
i k=1 k
5. Monoid and Group Rings.
Let R be a ring and S a monoid with identity e. The monoid ring
R[S] of S over R is the set of all formal finite sums
(cid:88)
f = r s,
s
s∈S
(so only finitely many r are non-zero) where each r ∈ R, s ∈ S.
s s
The set {s ∈ S | r (cid:54)= 0} is called the support of f, and is denoted
s
6 CHAPTER 1. GROUPS AND RINGS
supp(f). Note that, by definition, this set is always finite. Addition
and multiplication are defined as follows:
(cid:88) (cid:88) (cid:88)
a s+ b s = (a +b )s
s s s s
s∈S s∈S s∈S
and
(cid:88) (cid:88) (cid:88) (cid:88)
( a s)( b t) = ( a b )u,
s t s t
s∈S t∈S u∈S st=u
where the inner sum is evaluated over all s,t ∈ S in the respective
supports such that st = u. The elements of s ∈ S are identified with
1s; and the elements r ∈ R are identified with re. Clearly e = 1e is
the identity of the ring R[S].
In case S is a group, then R[S] is called a group ring. In case S
is the free abelian monoid with free basis X ,··· ,X then R[S] =
1 n
R[X ,··· ,X ]. If S is the free monoid in X ,··· ,X then R[S] is the
1 n 1 n
polynomial ring in noncommuting variables.
6. Contracted Semigroup Rings.
Let S be a semigroup with zero θ. The contracted semigroup ring of
S over a ring R is the ring R[S]/Rθ. This ring is denoted
R [S].
0
ThustheelementsofR [S]maybeidentifiedwiththesetoffinitesums
0
(cid:80)
r s with each r ∈ R, s ∈ S \{θ}, subject to the componentwise
s s
addition and multiplication defined on the R-basis S \{θ} as follows
(cid:26)
st if st (cid:54)= θ
s◦t =
0 if st = θ
and then extended by distributivity to all elements.
If S does not have a zero element, then we put R [S] = R[S].
0
Some important natural classes of rings may be treated as contracted
semigroup rings and not as semigroup rings. For example, let n > 1
be an integer, and let S be the semigroup of n×n matrix units, that
is,
S = {e | 1 ≤ i,j ≤ n}∪{θ}
ij
subject to the multiplication
(cid:26)
e if j = k
e e = il
ij kl θ if j (cid:54)= k
Then, for any ring R,
∼
R [S] = M (R),
0 n
1.2. RING CONSTRUCTIONS 7
a natural isomorphism. Now, if K is a field then M (K) is a simple
n
ring. Butasemigroupalgebraofanontrivialsemigroupisneversimple
as it contains the augmentation ideal, that is the ideal of all elements
of the semigroup algebra of which the sum of the coefficients is zero.
This example can be extended as follows. Let R be a K-algebra.
Let I,M be nonempty sets and P = (p ) a generalized M×I
mi m∈M,i∈I
matrixwithp ∈ R. ConsiderthesetM(R,I,M,P)ofallgeneralized
mi
I ×M matrices over R with finitely many nonzero entries. For any
A = (a ),B = (b ) ∈ M(R,I,M,P), addition and multiplication
im im
are defined as follows:
A+B = (c ) with c = a +b ,
im im im im
AB = A◦P ◦B where ◦ is the matrix multiplication
kA = (ka )
im
With these operations M(R,I,M,P) becomes a ring, called a ring
of matrix type. If each row and column of P contains an invertible
element of R then we call M(R,I,M,P) a Munn algebra.
Later we will see the notion of matrix semigroup M0(G0,I,M,P)
where G is a group, and P is a matrix with entries in G0. It is then
easily verified that
R [M0(G0,I,M,P)] ∼= M(R,I,M,P).
0
Denote by Mrow(R) the ring of all I×I matrices over R with finitely
I
manynonzerorows,subjecttothenaturaladditionandmultiplication.
The following is easily verified. The mapping
M(R,I,M,P) → Mrow(R) : A (cid:55)→ A◦P
I
is ring homomorphism.
Another useful class of contracted semigroup algebras are the mono-
mial algebras K[X]/I, where X is the free monoid on an alphabet
{x | i ∈ Ω} and I is an ideal generated by an ideal J of X. So,
i
K[X]/I is a K-space with basis the Rees factor semigroup X/J (the
∼
terminology will be explained later); thus K[X]/I = K [X/J].
0
7. Skew Polynomial Rings.
The notion of polynomial ring over a ring R in one variable X can be
extendedinsuchawaythatthevariabledoesnotnecessarilycommute
with the elements of R.
Let R be a ring and σ a ring homomorphism from R to R. The skew
polynomial ring R[X,σ] is, as in the case of polynomial rings, the set