ADVANCED ALGEBRA
Prof. Dr. B. Pareigis
Winter Semester 2001/02
Table of Contents
1. Tensor Products and Free Modules 3
1.1. Modules 3
1.2. Tensor products I 5
1.3. Free modules 6
1.4. Tensor products II 8
1.5. Bimodules 9
1.6. Complexes and exact sequences 12
2. Algebras and Coalgebras 15
2.1. Algebras 15
2.2. Tensor algebras 17
2.3. Symmetric algebras 19
2.4. Exterior algebras 21
2.5. Left A-modules 23
2.6. Coalgebras 23
2.7. Comodules 26
3. Projective Modules and Generators 30
3.1. Products and coproducts 30
3.2. Projective modules 34
3.3. Dual basis 36
3.4. Generators 39
4. Categories and Functors 40
4.1. Categories 40
4.2. Functors 42
4.3. Natural Transformations 43
5. Representable and Adjoint Functors, the Yoneda Lemma 46
,2 Advanced Algebra – Pareigis
5.1. Representable functors 46
5.2. The Yoneda Lemma 49
5.3. Adjoint functors 51
5.4. Universal problems 52
6. Limits and Colimits, Products and Equalizers 55
6.1. Limits of diagrams 55
6.2. Colimits of diagrams 57
6.3. Completeness 58
6.4. Adjoint functors and limits 59
7. The Morita Theorems 60
8. Simple and Semisimple rings and Modules 66
8.1. Simple and Semisimple rings 66
8.2. Injective Modules 67
8.3. Simple and Semisimple Modules 70
8.4. Noetherian Modules 73
9. Radical and Socle 76
10. Localization 81
10.1. Local rings 81
10.2. Localization 81
11. Monoidal Categories 87
12. Bialgebras and Hopf Algebras 92
12.1. Bialgebras 92
12.2. Hopf Algebras 94
13. Quickies in Advanced Algebra 101
, Tensor products and free modules 3
1. Tensor Products and Free Modules
1.1. Modules.
Definition 1.1. Let R be a ring (always associative with unit element). A left R-module R M
is an Abelian group M (with composition written as addition) together with an operation
R × M 3 (r, m) 7→ rm ∈ M
such that
(1) (rs)m = r(sm),
(2) (r + s)m = rm + sm,
(3) r(m + m0 ) = rm + rm0 ,
(4) 1m = m
for all r, s ∈ R, m, m0 ∈ M .
If R is a field then a (left) R-module is a (called a) vector space over R.
A homomorphism of left R-modules or simply an R-module homomorphism f : R M − → RN
is a homomorphism of groups with f (rm) = rf (m).
Right R-modules and homomorphisms of right R-modules are defined analogously.
We define
HomR (.M, .N ) := {f : R M −
→ R N |f is a homomorphism of left R-modules}.
Similarly HomR (M., N.) denotes the set of homomorphisms of right R-modules MR and NR .
An R-module homomorphism f : R M − → R N is
a monomorphism if f is injective,
an epimorphism if f is surjective,
an isomorphism if f is bijective,
an endomorphism if M = N ,
an automorphism if f is an endomorphism and an isomorphism.
Problem 1.1. Let R be a ring and M be an Abelian group. Show that there is a one-to-one
correspondence between maps f : R × M −
→ M that make M into a left R-module and ring
homomorphisms (always preserving the unit element) g : R −
→ End(M ).
Lemma 1.2. HomR (M, N ) is an Abelian group by (f + g)(m) := f (m) + g(m).
Proof. Since N is an Abelian group the set of maps Map(M, N ) is also an Abelian group.
The set of group homomorphisms Hom(M, N ) is a subgroup of Map(M, N ) (observe that this
holds only for Abelian groups). We show that HomR (M, N ) is a subgroup of Hom(M, N ).
We must only show that f − g is an R-module homomorphism if f and g are. Obviously
f − g is a group homomorphism. Furthermore we have (f − g)(rm) = f (rm) − g(rm) =
rf (m) − rg(m) = r(f (m) − g(m)) = r(f − g)(m).
Problem 1.2. Let f : M −→ N be an R-module homomorphism.
(1) f is an isomorphism if and only if (iff) there exists an R-module homomorphism
g:N − → M such that
f g = idN and gf = idM .
Furthermore g is uniquely determined by f .
(2) The following are equivalent:
(a) f is a monomorphism,
(b) for all R-modules P and all homomorphisms g, h : P −
→M
f g = f h =⇒ g = h,
, 4 Advanced Algebra – Pareigis
(c) for all R-modules P the homomorphism of Abelian groups
HomR (P, f ) : HomR (P, M ) 3 g 7→ f g ∈ HomR (P, N )
is a monomorphism.
(3) The following are equivalent:
(a) f is an epimorphism,
(b) for all R-modules P and all homomorphisms g, h : N −
→P
gf = hf =⇒ g = h,
(c) for all R-modules P the homomorphism of Abelian groups
HomR (f, P ) : HomR (N, P ) 3 g 7→ gf ∈ HomR (M, P )
is a monomorphism.
Remark 1.3. Each Abelian group is a Z-module in a unique way. Each homomorphism of
Abelian groups is a Z-module homomorphism.
Proof. By exercise 1.1 we have to find a unique ring homomorphism g : Z − → End(M ).
This holds more generally. If S is a ring then there is a unique ring homomorphism g : Z
−→ S. Since a ring homomorphism must preserve the unit we have g(1) = 1. Define
g(n) := 1 + . . . + 1 (n-times) for n ≥ 0 and g(−n) := −(1 + . . . + 1) (n-times) for n > 0. Then
it is easy to check that g is a ring homomorphism and it is obviously unique. This means
that M is a Z-module by nm = m+. . .+m (n-times) for n ≥ 0 and (−n)m = −(m+. . .+m)
(n-times) for n > 0.
If f : M − → N is a homomorphism of (Abelian) groups then f (nm) = f (m + . . . + m) =
f (m) + . . . + f (m) = nf (m) for n ≥ 0 and f ((−n)m) = f (−(m + . . . + m)) = −(f (m) +
. . . + f (m)) = (−n)f (m) for n > 0. Hence f is a Z-module homomorphism.
Problem 1.3. (1) Let R be a ring. Then R R is a left R-module.
(2) Let M be a Abelian group and End(M ) be the endomorphism ring of M . Then M
is an End(M )-module.
(3) {(1̄, 0̄), (0̄, 1̄)} is a generating set for the Z-module Z/(2) × Z/(3).
(4) {(1̄, 1̄)} is a generating set for the Z-module Z/(2) × Z/(3).
(5) Z Z/(n) has no basis as a module, i.e. this module is not free.
(6) Let V = ∞
L
i=0 Kbi be a countably infinite dimensional vector space over the field K.
Let p, q, a, b ∈ Hom(V, V ) be defined by
p(bi ) := b2i ,
q(bi ) := (
b2i+1 ,
bi/2 , if i is even, and
a(bi ) :=
0, if i is odd.
(
bi−1/2 , if i is odd, and
b(bi ) :=
0, if i is even.
Show pa + qb = idV , ap = bq = id, aq = bp = 0.
Show for R = EndK (V ) that R R = Ra ⊕ Rb and RR = pR ⊕ qR holds.
(7) Are {(0, . . . , a, . . . , 0)|a ∈ Kn } and {(a, 0, . . . , 0)|a ∈ Kn } isomorphic as Mn (K)-
modules?
(8) For each module P there is a module Q such that P ⊕ Q ∼ = Q.
(9) Which of the following statements is correct?
Prof. Dr. B. Pareigis
Winter Semester 2001/02
Table of Contents
1. Tensor Products and Free Modules 3
1.1. Modules 3
1.2. Tensor products I 5
1.3. Free modules 6
1.4. Tensor products II 8
1.5. Bimodules 9
1.6. Complexes and exact sequences 12
2. Algebras and Coalgebras 15
2.1. Algebras 15
2.2. Tensor algebras 17
2.3. Symmetric algebras 19
2.4. Exterior algebras 21
2.5. Left A-modules 23
2.6. Coalgebras 23
2.7. Comodules 26
3. Projective Modules and Generators 30
3.1. Products and coproducts 30
3.2. Projective modules 34
3.3. Dual basis 36
3.4. Generators 39
4. Categories and Functors 40
4.1. Categories 40
4.2. Functors 42
4.3. Natural Transformations 43
5. Representable and Adjoint Functors, the Yoneda Lemma 46
,2 Advanced Algebra – Pareigis
5.1. Representable functors 46
5.2. The Yoneda Lemma 49
5.3. Adjoint functors 51
5.4. Universal problems 52
6. Limits and Colimits, Products and Equalizers 55
6.1. Limits of diagrams 55
6.2. Colimits of diagrams 57
6.3. Completeness 58
6.4. Adjoint functors and limits 59
7. The Morita Theorems 60
8. Simple and Semisimple rings and Modules 66
8.1. Simple and Semisimple rings 66
8.2. Injective Modules 67
8.3. Simple and Semisimple Modules 70
8.4. Noetherian Modules 73
9. Radical and Socle 76
10. Localization 81
10.1. Local rings 81
10.2. Localization 81
11. Monoidal Categories 87
12. Bialgebras and Hopf Algebras 92
12.1. Bialgebras 92
12.2. Hopf Algebras 94
13. Quickies in Advanced Algebra 101
, Tensor products and free modules 3
1. Tensor Products and Free Modules
1.1. Modules.
Definition 1.1. Let R be a ring (always associative with unit element). A left R-module R M
is an Abelian group M (with composition written as addition) together with an operation
R × M 3 (r, m) 7→ rm ∈ M
such that
(1) (rs)m = r(sm),
(2) (r + s)m = rm + sm,
(3) r(m + m0 ) = rm + rm0 ,
(4) 1m = m
for all r, s ∈ R, m, m0 ∈ M .
If R is a field then a (left) R-module is a (called a) vector space over R.
A homomorphism of left R-modules or simply an R-module homomorphism f : R M − → RN
is a homomorphism of groups with f (rm) = rf (m).
Right R-modules and homomorphisms of right R-modules are defined analogously.
We define
HomR (.M, .N ) := {f : R M −
→ R N |f is a homomorphism of left R-modules}.
Similarly HomR (M., N.) denotes the set of homomorphisms of right R-modules MR and NR .
An R-module homomorphism f : R M − → R N is
a monomorphism if f is injective,
an epimorphism if f is surjective,
an isomorphism if f is bijective,
an endomorphism if M = N ,
an automorphism if f is an endomorphism and an isomorphism.
Problem 1.1. Let R be a ring and M be an Abelian group. Show that there is a one-to-one
correspondence between maps f : R × M −
→ M that make M into a left R-module and ring
homomorphisms (always preserving the unit element) g : R −
→ End(M ).
Lemma 1.2. HomR (M, N ) is an Abelian group by (f + g)(m) := f (m) + g(m).
Proof. Since N is an Abelian group the set of maps Map(M, N ) is also an Abelian group.
The set of group homomorphisms Hom(M, N ) is a subgroup of Map(M, N ) (observe that this
holds only for Abelian groups). We show that HomR (M, N ) is a subgroup of Hom(M, N ).
We must only show that f − g is an R-module homomorphism if f and g are. Obviously
f − g is a group homomorphism. Furthermore we have (f − g)(rm) = f (rm) − g(rm) =
rf (m) − rg(m) = r(f (m) − g(m)) = r(f − g)(m).
Problem 1.2. Let f : M −→ N be an R-module homomorphism.
(1) f is an isomorphism if and only if (iff) there exists an R-module homomorphism
g:N − → M such that
f g = idN and gf = idM .
Furthermore g is uniquely determined by f .
(2) The following are equivalent:
(a) f is a monomorphism,
(b) for all R-modules P and all homomorphisms g, h : P −
→M
f g = f h =⇒ g = h,
, 4 Advanced Algebra – Pareigis
(c) for all R-modules P the homomorphism of Abelian groups
HomR (P, f ) : HomR (P, M ) 3 g 7→ f g ∈ HomR (P, N )
is a monomorphism.
(3) The following are equivalent:
(a) f is an epimorphism,
(b) for all R-modules P and all homomorphisms g, h : N −
→P
gf = hf =⇒ g = h,
(c) for all R-modules P the homomorphism of Abelian groups
HomR (f, P ) : HomR (N, P ) 3 g 7→ gf ∈ HomR (M, P )
is a monomorphism.
Remark 1.3. Each Abelian group is a Z-module in a unique way. Each homomorphism of
Abelian groups is a Z-module homomorphism.
Proof. By exercise 1.1 we have to find a unique ring homomorphism g : Z − → End(M ).
This holds more generally. If S is a ring then there is a unique ring homomorphism g : Z
−→ S. Since a ring homomorphism must preserve the unit we have g(1) = 1. Define
g(n) := 1 + . . . + 1 (n-times) for n ≥ 0 and g(−n) := −(1 + . . . + 1) (n-times) for n > 0. Then
it is easy to check that g is a ring homomorphism and it is obviously unique. This means
that M is a Z-module by nm = m+. . .+m (n-times) for n ≥ 0 and (−n)m = −(m+. . .+m)
(n-times) for n > 0.
If f : M − → N is a homomorphism of (Abelian) groups then f (nm) = f (m + . . . + m) =
f (m) + . . . + f (m) = nf (m) for n ≥ 0 and f ((−n)m) = f (−(m + . . . + m)) = −(f (m) +
. . . + f (m)) = (−n)f (m) for n > 0. Hence f is a Z-module homomorphism.
Problem 1.3. (1) Let R be a ring. Then R R is a left R-module.
(2) Let M be a Abelian group and End(M ) be the endomorphism ring of M . Then M
is an End(M )-module.
(3) {(1̄, 0̄), (0̄, 1̄)} is a generating set for the Z-module Z/(2) × Z/(3).
(4) {(1̄, 1̄)} is a generating set for the Z-module Z/(2) × Z/(3).
(5) Z Z/(n) has no basis as a module, i.e. this module is not free.
(6) Let V = ∞
L
i=0 Kbi be a countably infinite dimensional vector space over the field K.
Let p, q, a, b ∈ Hom(V, V ) be defined by
p(bi ) := b2i ,
q(bi ) := (
b2i+1 ,
bi/2 , if i is even, and
a(bi ) :=
0, if i is odd.
(
bi−1/2 , if i is odd, and
b(bi ) :=
0, if i is even.
Show pa + qb = idV , ap = bq = id, aq = bp = 0.
Show for R = EndK (V ) that R R = Ra ⊕ Rb and RR = pR ⊕ qR holds.
(7) Are {(0, . . . , a, . . . , 0)|a ∈ Kn } and {(a, 0, . . . , 0)|a ∈ Kn } isomorphic as Mn (K)-
modules?
(8) For each module P there is a module Q such that P ⊕ Q ∼ = Q.
(9) Which of the following statements is correct?
