Algebras
Historically the first rings to be studied (in the second half of the 19th century) were
the rings of integers in algebraic number fields. At about the same time the theory of
algebras began to develop; its most important landmarks were the Wedderburn
structure theorems for semisimple algebras, and the study of the radical. The theories
merged when it was realized that the Wedderburn theorems could be stated more
generally for Artinian rings. This is the form in which the results will be presented
here, in Section 5.2 and 5.3; the formulation for general rings (Jacobson radical
and density theorem) will be reserved for FA. As a preparation for this study we
examine the form the tensor product takes for algebras in Section 5.4, and in Section
5.5 we introduce scalar invariants. In Section 5.6 algebras are used to define an
important number-theoretic function, the Mobius function.
5.1 Algebras; Definition and Examples
Most of our rings have a coefficient ring; this is often given as a field, so that the ring
under consideration is a vector space. But it is convenient to frame the definitions
more generally; all we need to impose on the coefficient ring is commutativity. At
a later stage we shall see how even this restriction can be lifted.
Thus let K be a commutative ring; by an algebra over K or K-algebra we under-
stand a ring A which is also a K-module, such that the multiplication in A is bilinear.
Explicitly we have in addition to the ring laws,
ax.y = x.ay = a(xy), for any x,y E A, a E K. (5.1.1)
For example, K itself is always a K-algebra in a natural way. We also note that any
ring R may be regarded as a Z-algebra by putting
na = a + a + ... + a(n terms), ( - n)a = -na, a E R, n E N. (5.1.2)
It is easily verified that R becomes a Z-algebra in this way.
A homomorphism of K-algebras is a ring homomorphism which is K-linear, and a
subalgebra of a K-algebra A is a subring admitting multiplication by all the elements
of K. For example, the centre of A, defined as C = {z E Alzx = xz for all x E A} is a
subalgebra of A, as is easily checked.
131
,132 Algebras
From (5.1.1) we find by taking y = 1 that x.a1 = ax, while x = 1 gives a1.y = ay.
Therefore
a1 .x=x.a1=ax, xEA,aEK.
Taking x = fJ1, we obtain al.fJ1 = a(fJl) = afJ.1. These rules, together with the
equations (a + fJ).l = a1 + fJ1 and 1.1 = I, which hold in any K-module, show
that the map a 1---+ a1 is a homomorphism of K into the centre of A. Conversely,
given any ring R and a homomorphism f from K to the centre of R, we can define
R as a K-algebra by putting
ax = (af)x, x E R, a E K.
This shows a K-algebra A to be nothing more than a ring with a homomorphism
from K to the centre of A. In particular, any ring whose centre contains K may be
regarded as a K-algebra.
Sometimes a K-algebra is defined more generally as a K-module A with a bilinear
multiplication. If A contains an element 1 such that x . 1 = 1. x = x for all x E A, the
algebra A is said to be unital. What we described above is the special case when A
is unital and associative, and so is a ring. Occasionally we shall consider the term
'algebra' in this more general sense, but when nothing is said to the contrary,
K-algebras are understood to be associative and unital.
We remark that for a non-unital K-algebra, i.e. lacking a I, we can always adjoin a
1 by forming the direct sum K EEl A with the multiplication
(a, a)(fJ, b) = (afJ, ab + fJa + ab), a, b E A, a, fJ E K. (5.1.3)
The resulting K-algebra is denoted by AI; it has the one (1,0) and is associative if
A is.
A K-algebra A is said to be augmented if there is an algebra homomorphism
c : A ---+ K. The kernel of c is an ideal of A, called the augmentation ideal. Augmented
algebras may be described as follows:
Theorem 5.1.1. A K-algebra A is augmented if and only if the homomorphism
a 1---+ a1 (a E K) is an embedding and A contains an ideal a such that
A=aEElK.1. (5.1.4)
Proof. Assume that A is an augmented K-algebra and write a = ker c for the aug-
mentation ideal. Any x E A may be written in the form
x = (x -xc) +xc,
where xc EK and (x-xc)c=xc-xc=O, hence a+K.1 =A, and anK.1 =0,
because c restricted to K is injective, so we have the direct sum (5.1.4).
Conversely, if A contains an ideal a and a copy K.1 of K such that (5.1.4) holds,
then the projection on K.1 is an augmentation, as is easily verified. •
,5.1 Algebras; Definition and Examples 133
We observe that for any algebra A without a 1 the algebra AI, formed as in (5.1.3),
is augmented. Conversely, given an augmented algebra as in (5.1.4), we have A = a l ;
the verification is straightforward and may be left to the reader.
If our coefficient ring is a field k, any k-algebra A is a vector space over k and so
has a basis {u;} say. Every element of A can then be uniquely expressed in the form
L aiUi (ai E k, almost all 0). In particular, the multiplication in A is described
completely by the products UiUj' These take the form
UiUj = L Yijr U" Yijr E k.
The elements Yijr are called the structure constants of A. They determine the algebra
structure completely, by bilinearity, but it must be remembered that (i) the Yijr
depend on the choice of the basis in A, and (ii) the Yijr cannot be assigned arbitrarily,
for they will need to satisfy the equations ensuring that A is associative and unital.
If every non-zero element has an inverse (and 1 =j:. 0), A is a skew field, which is
also called a division algebra when A is finite-dimensional over k.
Let us give some examples of K-algebras, where K is any commutative ring.
Example 1. Let Vbe any K-module and put A = EndK(V). We can regard A as a K-
module by defining for f E A, a E K, x E V, af : x 1--+ a(x!) = (ax)f. With this
definition it is easily checked that A is indeed a K-algebra. In particular, when V
is the free K-module of rank n, V = Kn, then A ~ 9Jln(K), the full n x n matrix
ring over K. This is easily verified directly, but follows also from Theorem 5.1.3
below and Corollary 4.4.2.
Example 2. 'In (K), the set of all upper triangular n x n matrices over K, i.e. matrices
C = (Cij), where Cij = 0 for i > j.
Example 3. Let A be a finite-dimensional vector space over a field k, with basis
UI, ... , Un say. Then for any elements Cijk E k, a k-algebra on A (not necessarily asso-
ciative or commutative, or with a one) is defined by setting UiUj = L CijkUk. Thus A
may be defined by its 'multiplication table'.
Example 4. Let M = {uili E J} be a monoid and take A to be the free K-module on M
as basis, with multiplication UiUj = Uk as in M. By linearity this defines a multi-
plication on A and it is not hard to see that A becomes a K-algebra in this way.
The associativity of A follows from the associativity of M, and the unit element of
M is also the 1 for A. This algebra is usually denoted by KM and is called the
monoid algebra on M; in particular, for a group G we obtain the group algebra
KG. The mapping 8 : L aiUi 1--+ L ai is an augmentation for KM. The following
are particular cases of this construction.
(i) Let M = {I, x, Xl, ... }consist of all powers of a single element x. Then M is the
infinite cyclic monoid and KM consists of all polynomials in x over K, with the
usual multiplication, thus KM ~ K[x].
(ii) Let G be the infinite cyclic group, with generator x. The group algebra is
K[x, X-I], the ring of all Laurent polynomials in the indeterminate x.
, 134 Algebras
(iii) Let G = Cn be the cyclic group of order n, with generator u. Then KC n consists
of all 'polynomials' ao + al u + ... + an _ I un - I with the usual multiplication,
but subject to un = 1.
(iv) Let M be the free monoid on XI, ... ,X" i.e. the set of all 'words' Xii Xi, ... Xi"
(1 ::: iv ::: r), with juxtaposition as multiplication, including the empty word
as the neutral element. The monoid algebra on K is called the free associative
algebra on XI, ... , Xr as free generating set and is denoted by K(x I' . .. , xr}.
Example 5. Any subring of C, the complex numbers, contains Z and so is a Z-
algebra. Given a E C, consider Z[a], the subring generated by a. As abelian group
this is not usually finitely generated; it is so precisely when a satisfies an equation
(5.1.5)
For if a satisfies (5.1.5), then we can express all powers of a as linear combinations of
1, a, a 2 , •. • , an-I, so Z[a] is the finitely generated as abelian group. The converse
also holds, as is not hard to verify (see Section 9.4). An equation or polynomial
with highest coefficient 1 is called monic, and a complex number satisfying a
monic equation (5.1.5) with integer coefficients is called an algebraic integer. Since
the conjugates satisfy the same equation, they are again algebraic integers, though
they need not lie in Z[a]. For example, Z[i], where i is a root of x2 + 1 = 0, is a
ring, the ring of Gaussian integers.
Example 6. Finite-dimensional algebras over a field. Let k be a field and A be an
n-dimensional k-algebra with basis UI, ... ,Un' As we have seen, A is completely
determined by the n3 structure constants Yijr' If we choose the basis so that
UI = 1, then
Ylir = Yilr = Oir,
while the associativity is expressed by the equations
L YijvYvrs = L YivsYjrv.
v v
Example 7. Any Boolean algebra may be regarded as an Fralgebra, as we saw in
Section 4.1.
Let A be an n-dimensional algebra over a field k and consider the right multiplica-
tion in A,
Pa : X I~ xa, X E A. (5.1.6)
This is a k-linear mapping and hence can be represented by an n x n matrix over k.
Bya matrix representation, or simply a representation, one understands a k-algebra
homomorphism into a full matrix ring over k. Now the right multiplication Pa is
a representation of A, as is easily checked. It is called the regular representation of
A; more precisely it is the right regular representation, the left regular representation
Historically the first rings to be studied (in the second half of the 19th century) were
the rings of integers in algebraic number fields. At about the same time the theory of
algebras began to develop; its most important landmarks were the Wedderburn
structure theorems for semisimple algebras, and the study of the radical. The theories
merged when it was realized that the Wedderburn theorems could be stated more
generally for Artinian rings. This is the form in which the results will be presented
here, in Section 5.2 and 5.3; the formulation for general rings (Jacobson radical
and density theorem) will be reserved for FA. As a preparation for this study we
examine the form the tensor product takes for algebras in Section 5.4, and in Section
5.5 we introduce scalar invariants. In Section 5.6 algebras are used to define an
important number-theoretic function, the Mobius function.
5.1 Algebras; Definition and Examples
Most of our rings have a coefficient ring; this is often given as a field, so that the ring
under consideration is a vector space. But it is convenient to frame the definitions
more generally; all we need to impose on the coefficient ring is commutativity. At
a later stage we shall see how even this restriction can be lifted.
Thus let K be a commutative ring; by an algebra over K or K-algebra we under-
stand a ring A which is also a K-module, such that the multiplication in A is bilinear.
Explicitly we have in addition to the ring laws,
ax.y = x.ay = a(xy), for any x,y E A, a E K. (5.1.1)
For example, K itself is always a K-algebra in a natural way. We also note that any
ring R may be regarded as a Z-algebra by putting
na = a + a + ... + a(n terms), ( - n)a = -na, a E R, n E N. (5.1.2)
It is easily verified that R becomes a Z-algebra in this way.
A homomorphism of K-algebras is a ring homomorphism which is K-linear, and a
subalgebra of a K-algebra A is a subring admitting multiplication by all the elements
of K. For example, the centre of A, defined as C = {z E Alzx = xz for all x E A} is a
subalgebra of A, as is easily checked.
131
,132 Algebras
From (5.1.1) we find by taking y = 1 that x.a1 = ax, while x = 1 gives a1.y = ay.
Therefore
a1 .x=x.a1=ax, xEA,aEK.
Taking x = fJ1, we obtain al.fJ1 = a(fJl) = afJ.1. These rules, together with the
equations (a + fJ).l = a1 + fJ1 and 1.1 = I, which hold in any K-module, show
that the map a 1---+ a1 is a homomorphism of K into the centre of A. Conversely,
given any ring R and a homomorphism f from K to the centre of R, we can define
R as a K-algebra by putting
ax = (af)x, x E R, a E K.
This shows a K-algebra A to be nothing more than a ring with a homomorphism
from K to the centre of A. In particular, any ring whose centre contains K may be
regarded as a K-algebra.
Sometimes a K-algebra is defined more generally as a K-module A with a bilinear
multiplication. If A contains an element 1 such that x . 1 = 1. x = x for all x E A, the
algebra A is said to be unital. What we described above is the special case when A
is unital and associative, and so is a ring. Occasionally we shall consider the term
'algebra' in this more general sense, but when nothing is said to the contrary,
K-algebras are understood to be associative and unital.
We remark that for a non-unital K-algebra, i.e. lacking a I, we can always adjoin a
1 by forming the direct sum K EEl A with the multiplication
(a, a)(fJ, b) = (afJ, ab + fJa + ab), a, b E A, a, fJ E K. (5.1.3)
The resulting K-algebra is denoted by AI; it has the one (1,0) and is associative if
A is.
A K-algebra A is said to be augmented if there is an algebra homomorphism
c : A ---+ K. The kernel of c is an ideal of A, called the augmentation ideal. Augmented
algebras may be described as follows:
Theorem 5.1.1. A K-algebra A is augmented if and only if the homomorphism
a 1---+ a1 (a E K) is an embedding and A contains an ideal a such that
A=aEElK.1. (5.1.4)
Proof. Assume that A is an augmented K-algebra and write a = ker c for the aug-
mentation ideal. Any x E A may be written in the form
x = (x -xc) +xc,
where xc EK and (x-xc)c=xc-xc=O, hence a+K.1 =A, and anK.1 =0,
because c restricted to K is injective, so we have the direct sum (5.1.4).
Conversely, if A contains an ideal a and a copy K.1 of K such that (5.1.4) holds,
then the projection on K.1 is an augmentation, as is easily verified. •
,5.1 Algebras; Definition and Examples 133
We observe that for any algebra A without a 1 the algebra AI, formed as in (5.1.3),
is augmented. Conversely, given an augmented algebra as in (5.1.4), we have A = a l ;
the verification is straightforward and may be left to the reader.
If our coefficient ring is a field k, any k-algebra A is a vector space over k and so
has a basis {u;} say. Every element of A can then be uniquely expressed in the form
L aiUi (ai E k, almost all 0). In particular, the multiplication in A is described
completely by the products UiUj' These take the form
UiUj = L Yijr U" Yijr E k.
The elements Yijr are called the structure constants of A. They determine the algebra
structure completely, by bilinearity, but it must be remembered that (i) the Yijr
depend on the choice of the basis in A, and (ii) the Yijr cannot be assigned arbitrarily,
for they will need to satisfy the equations ensuring that A is associative and unital.
If every non-zero element has an inverse (and 1 =j:. 0), A is a skew field, which is
also called a division algebra when A is finite-dimensional over k.
Let us give some examples of K-algebras, where K is any commutative ring.
Example 1. Let Vbe any K-module and put A = EndK(V). We can regard A as a K-
module by defining for f E A, a E K, x E V, af : x 1--+ a(x!) = (ax)f. With this
definition it is easily checked that A is indeed a K-algebra. In particular, when V
is the free K-module of rank n, V = Kn, then A ~ 9Jln(K), the full n x n matrix
ring over K. This is easily verified directly, but follows also from Theorem 5.1.3
below and Corollary 4.4.2.
Example 2. 'In (K), the set of all upper triangular n x n matrices over K, i.e. matrices
C = (Cij), where Cij = 0 for i > j.
Example 3. Let A be a finite-dimensional vector space over a field k, with basis
UI, ... , Un say. Then for any elements Cijk E k, a k-algebra on A (not necessarily asso-
ciative or commutative, or with a one) is defined by setting UiUj = L CijkUk. Thus A
may be defined by its 'multiplication table'.
Example 4. Let M = {uili E J} be a monoid and take A to be the free K-module on M
as basis, with multiplication UiUj = Uk as in M. By linearity this defines a multi-
plication on A and it is not hard to see that A becomes a K-algebra in this way.
The associativity of A follows from the associativity of M, and the unit element of
M is also the 1 for A. This algebra is usually denoted by KM and is called the
monoid algebra on M; in particular, for a group G we obtain the group algebra
KG. The mapping 8 : L aiUi 1--+ L ai is an augmentation for KM. The following
are particular cases of this construction.
(i) Let M = {I, x, Xl, ... }consist of all powers of a single element x. Then M is the
infinite cyclic monoid and KM consists of all polynomials in x over K, with the
usual multiplication, thus KM ~ K[x].
(ii) Let G be the infinite cyclic group, with generator x. The group algebra is
K[x, X-I], the ring of all Laurent polynomials in the indeterminate x.
, 134 Algebras
(iii) Let G = Cn be the cyclic group of order n, with generator u. Then KC n consists
of all 'polynomials' ao + al u + ... + an _ I un - I with the usual multiplication,
but subject to un = 1.
(iv) Let M be the free monoid on XI, ... ,X" i.e. the set of all 'words' Xii Xi, ... Xi"
(1 ::: iv ::: r), with juxtaposition as multiplication, including the empty word
as the neutral element. The monoid algebra on K is called the free associative
algebra on XI, ... , Xr as free generating set and is denoted by K(x I' . .. , xr}.
Example 5. Any subring of C, the complex numbers, contains Z and so is a Z-
algebra. Given a E C, consider Z[a], the subring generated by a. As abelian group
this is not usually finitely generated; it is so precisely when a satisfies an equation
(5.1.5)
For if a satisfies (5.1.5), then we can express all powers of a as linear combinations of
1, a, a 2 , •. • , an-I, so Z[a] is the finitely generated as abelian group. The converse
also holds, as is not hard to verify (see Section 9.4). An equation or polynomial
with highest coefficient 1 is called monic, and a complex number satisfying a
monic equation (5.1.5) with integer coefficients is called an algebraic integer. Since
the conjugates satisfy the same equation, they are again algebraic integers, though
they need not lie in Z[a]. For example, Z[i], where i is a root of x2 + 1 = 0, is a
ring, the ring of Gaussian integers.
Example 6. Finite-dimensional algebras over a field. Let k be a field and A be an
n-dimensional k-algebra with basis UI, ... ,Un' As we have seen, A is completely
determined by the n3 structure constants Yijr' If we choose the basis so that
UI = 1, then
Ylir = Yilr = Oir,
while the associativity is expressed by the equations
L YijvYvrs = L YivsYjrv.
v v
Example 7. Any Boolean algebra may be regarded as an Fralgebra, as we saw in
Section 4.1.
Let A be an n-dimensional algebra over a field k and consider the right multiplica-
tion in A,
Pa : X I~ xa, X E A. (5.1.6)
This is a k-linear mapping and hence can be represented by an n x n matrix over k.
Bya matrix representation, or simply a representation, one understands a k-algebra
homomorphism into a full matrix ring over k. Now the right multiplication Pa is
a representation of A, as is easily checked. It is called the regular representation of
A; more precisely it is the right regular representation, the left regular representation
