Rings and Modules
=====
Linear algebra deals with fields and vector spaces; here we are concerned with the
generalizations to rings and modules over them. Whereas a vector space over a
given field is determined up to isomorphism by its dimension, there is much greater
variety for modules. Another way of regarding modules is as abelian groups, written
additively, with operators. This means that much of general group theory applies,
and after recalling the isomorphism theorems, proved for groups in Section 2.3,
we treat a number of special situations. Semisimple modules (Section 4.3) come
closest to vector spaces; they are direct sums of simple modules, but we have to
bear in mind that over a given ring there may be more than one type of simple
module. In the free modules (Section 4.6) we have another generalization of
vector spaces. The homological treatment of module theory requires the notions
of projective and injective module (Section 4.7), and they can usefully be introduced
here, as they will occur again in Chapter 10, although their main use will be in FA.
Other important notions introduced here are those of matrix ring (Section 4.4) and
tensor product (Section 4.8).
4.1 The Definitions Recalled
We recall that a ring is a set R with two binary operations with values in R, addition:
x + y, and multiplication: xy, such that R is an abelian group under addition with
neutral element 0, the zero element, a monoid under multiplication with neutral
element 1, called the unit element or one, and these operations are linked by the
distributive laws:
x(y+z)=xy+xz, (x+y)z=xz+yz forallx,y,zER.
If 1 = 0, then for any x E R, x = x.l = x.o = 0, so the ring consists of a single
element. This is the trivial ring; usually our rings are assumed to be non-trivial. If
the commutative law holds: xy = yx for all x, y E R, the ring is said to be commuta-
tive, but we shall not generally make this assumption.
The best-known example of a ring is the ring of integers Z, which is in fact com-
mutative. The integers also have the property that a product is zero iff at least one of
the factors is zero. For any element c of a ring R either (i) c = 0, or (ii) c -::j:. 0 and
ca = 0 or ac = 0 for some a -::j:. 0 in R, or (iii) ac and ca never vanish for a -::j:. O.
79
,80 Rings and Modules
If (ii) holds, c is called a zerodivisor; if (iii) holds, c is a non-zerodivisor. A non-trivial
ring without zerodivisors is called an integral domain. Some authors use this term
only for commutative rings, but we shall not make this restriction. Thus a ring R
is an integral domain iff R\{O} contains 1 and is closed under multiplication; we
shall usually write R\{O} = RX. In an integral domain R the additive order of 1 is
called the characteristic of R. If n.l :f=. 0 for all n > 0, R is said to have characteristic
zero. The alternative is that 1 has finite order, which is easily seen to be a prime; thus
every integral domain has characteristic zero or a prime number.
An integral domain R for which R X is a group under multiplication is called a
division ring, or when R is commutative, a field. Instead of 'division ring' the term
skew field will often be used. Generally an element a E R is called a unit or invertible
if it has an inverse, i.e. there exists a-I E R such that aa - 1 = a-I a = 1. Clearly any
a E R can only have one inverse, for if a', a" are two inverses of a, then
a' = a'aa" = a". A non-unit is an element which is not a unit; however, an excep-
tion will be made for the 'matrix units' eij (i,j = 1, ... , n) to be introduced in
Section 4.4, which are of course non-units when n > 1.
Two rings R, S are said to be isomorphic, if there is a mapping f : R --+ S which is
a bijection preserving all the ring operations, i.e.
(x + y)f = xf + yf, (4.1.1)
(xy)f = xf.yf, (4.1.2)
If = 1, (4.1.3)
for all x, Y E R. More generally, a mapping f : R --+ S satisfying (4.1.1 )-( 4.1.3) is
called a homomorphism; such a mapping need not be a bijection. We note that for
any such homomorphism the image im f is a subring of S, i.e. a subset of S admitting
the operations of S, while the kernel of f, ker f = Of - 1, is an ideal of R, i.e. a
subgroup of the additive group of R such that R(ker f) ~ ker f, (ker f)R ~ ker f.
It is worth noting that for an isomorphism the condition (4.1.3) holds automatically,
since the 1 of S is uniquely determined as the solution of xa = ax = a for all a E S;
however, for a homomorphism it has to be postulated separately. As in the case of
groups we define an endomorphism of a ring as a homomorphism into itself and
an automorphism as an isomorphism with itself.
In a commutative ring A the ideal generated by a single element c is denoted by (c)
and is called a principal ideal. If A is a commutative integral domain in which every
ideal is principal, A is called a principal ideal domain, often abbreviated as PID.
In any ring R the set 81(R) of all central idempotents of R, i.e. elements e in the
centre of R such that t? = e, is a Boolean algebra under the operations
x /\ Y = xy, xV y =x +y - xy, x' =1- x. (4.1.4)
Shortly we shall see that every Boolean algebra may be obtained as the algebra of
idempotents of some commutative ring R; in fact R may be chosen to consist entirely
of idempotents. Let us define a Boolean ring as a ring satisfying the identity
(4.1.5)
,4.1 The Definitions Recalled 81
Such a ring is necessarily commutative. For we have 2x = (2X)2 = 4x2 = 4x, hence
2x = 0 and so
x=-x. ( 4.1.6)
Next we have x + y = (x + y)2 = x 2 + xy + yx + y2; hence xy + yx = 0, so by
(4.1.6), xy = yx. Thus every Boolean ring is commutative and of characteristic 2.
Proposition 4.1.1. (i) Given any ring R, the set 8l(R) of its central idempotents forms a
Boolean algebra relative to the operations (4.1.4).
(ii) On any Boolean algebra B define the two operations
x + y = (x /\ y') V (x' /\ y), xy = x /\ y. (4.1.7)
Then B forms a Boolean ring 9l(B) relative to these operations.
Moreover, !?4(91(B)) ~ B for any Boolean algebra Band PA(!?4(R)) ~ R for any
Boolean ring R.
The first operation (4.1.7) is called the symmetric difference of x and y; in the case
where B = &>(X), Y + Z represents the subset of X consisting of all elements that are
either in Y or in Z but not in both (this is also called 'addition mod 2').
Proof. The verification that 8l(R) is a Boolean algebra under the operations (4.1.4)
and that PA(B) is a ring under the operations (4.1.7) is routine and may be left to the
reader.
Now let B be a Boolean algebra and put Bl = 8l(PA(B)). Since 9l(B) is a Boolean
ring, Bl and B are the same set, and both are Boolean algebras with the same defini-
tion of I\. In Bl we have x' = 1 - x; from (4.1.7) and the fact that 9l(B) is a Boolean
ring we see that 1 - x = (1 /\ x') V (0/\ x) = x'. SO x' is the same on Band B 1; like-
wise for x /\ y, hence this also holds for x V y = (x' /\ y')'.
Next take a Boolean ring R and put Rl = PA(81(R)). Again Rand Rl are the same
set; both are Boolean rings with the same definition of product, and if +1 denotes the
sum in Rb then we have x +1 y = x(I - y) + (1 - x)y = x + y; hence Rl ~ R, as
required. •
It is clear that Boolean algebra homomorphisms correspond precisely to ring
homomorphisms; we have here an example of a category equivalence: PA and !?4
are mutually inverse functors defining an equivalence between the categories of
Boolean rings and Boolean algebras.
Just as a group can act by permutations on a set, so a ring can act by linear trans-
formations on an additive group. This gives rise to the notion of a module, which
may also be regarded as a generalization of a vector space. Thus let R be any ring.
Bya right R-module one understands an abelian group M, written additively, with
a mapping from M x R into M, (x, a) 1--+ xa, such that for all x, y E M, a, b E R,
(x + y)a = xa + ya, (4.1.8)
x(a+b) =xa+xb, (4.1.9)
, 82 Rings and Modules
x(ab) = (xa)b, (4.1.10)
xl =x. (4.1.11)
Sometimes we have, instead of (4.1.10), the rule x(ab) = (xb)a; in such cases it is
expedient to write the operator on the left, putting ax instead of xa, so this law
now takes the form
(ab)x = a(bx). (4.1.12)
The resulting structure is called a left R-module. Ifwe define the opposite ring of R as
the ring RO whose additive group is the same as that of R but with the multiplication
x.y = yx (x,y E R)
then we can say: a left R-module is a right RO-module and a right R-module is a left
RO -module. In other words, changing the side from which a ring operates on a
module corresponds to a reversal of the order of multiplication in the ring. This
means that instead of using an RO-module we can take the coefficients on the
other side so as to get an R-module. Of course when R is commutative then
RO = R; in this case the difference between left and right R-modules is purely nota-
tional. We shall sometimes write RM resp. MR to indicate that M is a left resp. right
R-module. In general we shall speak of an 'R-module' when the side is not specified,
and a 'module' when we do not wish to specify the ring. A module is called cyclic if it
can be generated by a single element.
As an example we may take the ring itself. Any ring R is a right R-module, with the
action of R being defined by right multiplication. The distributive laws ensure that
(4.1.8), (4.1.9) hold while (4.1.10) follows by the associative law and (4.1.11) is
the property of the unit element. In fact R as right (or left) R-module is cyclic,
generated by 1. Any submodule, i.e. subgroup admitting the right multiplication
by elements of R, is called a right ideal of R. Similarly, left multiplication defines
R as a left R-module, with left ideals as submodules; the ideals defined earlier are
just subsets of R that are both left ideals and right ideals, also called two-sided ideals.
Another example of importance in the sequel is obtained by taking any abelian
group A, written additively, and considering the set E = End(A) of all its endo-
morphisms. If a, bare endomorphisms acting on the right, and x, YEA, then
(4.1.8) just expresses their defining property. We can now define an addition on E
by (4.1.9) and a multiplication by (4.1.10), and can verify that E forms a ring for
these operations, with the identity mapping as unit element. This verification is
straightforward and so may be left to the reader. In this way we obtain a ring struc-
ture on E, called the endomorphism ring of A, and A is seen to be a right E-module.
Given two right R-modules M, N over a ring R, an R-homomorphism f : M -+ N,
also called an R-linear map, is defined as a map f satisfying
f(x + y) = fx + fy, f(xa) = (fx)a for all x, y E M, a E R.
Here we have written f on the left, on the opposite side from the ring coefficients.
We shall usually follow this practice; thus a homomorphism of left modules will
=====
Linear algebra deals with fields and vector spaces; here we are concerned with the
generalizations to rings and modules over them. Whereas a vector space over a
given field is determined up to isomorphism by its dimension, there is much greater
variety for modules. Another way of regarding modules is as abelian groups, written
additively, with operators. This means that much of general group theory applies,
and after recalling the isomorphism theorems, proved for groups in Section 2.3,
we treat a number of special situations. Semisimple modules (Section 4.3) come
closest to vector spaces; they are direct sums of simple modules, but we have to
bear in mind that over a given ring there may be more than one type of simple
module. In the free modules (Section 4.6) we have another generalization of
vector spaces. The homological treatment of module theory requires the notions
of projective and injective module (Section 4.7), and they can usefully be introduced
here, as they will occur again in Chapter 10, although their main use will be in FA.
Other important notions introduced here are those of matrix ring (Section 4.4) and
tensor product (Section 4.8).
4.1 The Definitions Recalled
We recall that a ring is a set R with two binary operations with values in R, addition:
x + y, and multiplication: xy, such that R is an abelian group under addition with
neutral element 0, the zero element, a monoid under multiplication with neutral
element 1, called the unit element or one, and these operations are linked by the
distributive laws:
x(y+z)=xy+xz, (x+y)z=xz+yz forallx,y,zER.
If 1 = 0, then for any x E R, x = x.l = x.o = 0, so the ring consists of a single
element. This is the trivial ring; usually our rings are assumed to be non-trivial. If
the commutative law holds: xy = yx for all x, y E R, the ring is said to be commuta-
tive, but we shall not generally make this assumption.
The best-known example of a ring is the ring of integers Z, which is in fact com-
mutative. The integers also have the property that a product is zero iff at least one of
the factors is zero. For any element c of a ring R either (i) c = 0, or (ii) c -::j:. 0 and
ca = 0 or ac = 0 for some a -::j:. 0 in R, or (iii) ac and ca never vanish for a -::j:. O.
79
,80 Rings and Modules
If (ii) holds, c is called a zerodivisor; if (iii) holds, c is a non-zerodivisor. A non-trivial
ring without zerodivisors is called an integral domain. Some authors use this term
only for commutative rings, but we shall not make this restriction. Thus a ring R
is an integral domain iff R\{O} contains 1 and is closed under multiplication; we
shall usually write R\{O} = RX. In an integral domain R the additive order of 1 is
called the characteristic of R. If n.l :f=. 0 for all n > 0, R is said to have characteristic
zero. The alternative is that 1 has finite order, which is easily seen to be a prime; thus
every integral domain has characteristic zero or a prime number.
An integral domain R for which R X is a group under multiplication is called a
division ring, or when R is commutative, a field. Instead of 'division ring' the term
skew field will often be used. Generally an element a E R is called a unit or invertible
if it has an inverse, i.e. there exists a-I E R such that aa - 1 = a-I a = 1. Clearly any
a E R can only have one inverse, for if a', a" are two inverses of a, then
a' = a'aa" = a". A non-unit is an element which is not a unit; however, an excep-
tion will be made for the 'matrix units' eij (i,j = 1, ... , n) to be introduced in
Section 4.4, which are of course non-units when n > 1.
Two rings R, S are said to be isomorphic, if there is a mapping f : R --+ S which is
a bijection preserving all the ring operations, i.e.
(x + y)f = xf + yf, (4.1.1)
(xy)f = xf.yf, (4.1.2)
If = 1, (4.1.3)
for all x, Y E R. More generally, a mapping f : R --+ S satisfying (4.1.1 )-( 4.1.3) is
called a homomorphism; such a mapping need not be a bijection. We note that for
any such homomorphism the image im f is a subring of S, i.e. a subset of S admitting
the operations of S, while the kernel of f, ker f = Of - 1, is an ideal of R, i.e. a
subgroup of the additive group of R such that R(ker f) ~ ker f, (ker f)R ~ ker f.
It is worth noting that for an isomorphism the condition (4.1.3) holds automatically,
since the 1 of S is uniquely determined as the solution of xa = ax = a for all a E S;
however, for a homomorphism it has to be postulated separately. As in the case of
groups we define an endomorphism of a ring as a homomorphism into itself and
an automorphism as an isomorphism with itself.
In a commutative ring A the ideal generated by a single element c is denoted by (c)
and is called a principal ideal. If A is a commutative integral domain in which every
ideal is principal, A is called a principal ideal domain, often abbreviated as PID.
In any ring R the set 81(R) of all central idempotents of R, i.e. elements e in the
centre of R such that t? = e, is a Boolean algebra under the operations
x /\ Y = xy, xV y =x +y - xy, x' =1- x. (4.1.4)
Shortly we shall see that every Boolean algebra may be obtained as the algebra of
idempotents of some commutative ring R; in fact R may be chosen to consist entirely
of idempotents. Let us define a Boolean ring as a ring satisfying the identity
(4.1.5)
,4.1 The Definitions Recalled 81
Such a ring is necessarily commutative. For we have 2x = (2X)2 = 4x2 = 4x, hence
2x = 0 and so
x=-x. ( 4.1.6)
Next we have x + y = (x + y)2 = x 2 + xy + yx + y2; hence xy + yx = 0, so by
(4.1.6), xy = yx. Thus every Boolean ring is commutative and of characteristic 2.
Proposition 4.1.1. (i) Given any ring R, the set 8l(R) of its central idempotents forms a
Boolean algebra relative to the operations (4.1.4).
(ii) On any Boolean algebra B define the two operations
x + y = (x /\ y') V (x' /\ y), xy = x /\ y. (4.1.7)
Then B forms a Boolean ring 9l(B) relative to these operations.
Moreover, !?4(91(B)) ~ B for any Boolean algebra Band PA(!?4(R)) ~ R for any
Boolean ring R.
The first operation (4.1.7) is called the symmetric difference of x and y; in the case
where B = &>(X), Y + Z represents the subset of X consisting of all elements that are
either in Y or in Z but not in both (this is also called 'addition mod 2').
Proof. The verification that 8l(R) is a Boolean algebra under the operations (4.1.4)
and that PA(B) is a ring under the operations (4.1.7) is routine and may be left to the
reader.
Now let B be a Boolean algebra and put Bl = 8l(PA(B)). Since 9l(B) is a Boolean
ring, Bl and B are the same set, and both are Boolean algebras with the same defini-
tion of I\. In Bl we have x' = 1 - x; from (4.1.7) and the fact that 9l(B) is a Boolean
ring we see that 1 - x = (1 /\ x') V (0/\ x) = x'. SO x' is the same on Band B 1; like-
wise for x /\ y, hence this also holds for x V y = (x' /\ y')'.
Next take a Boolean ring R and put Rl = PA(81(R)). Again Rand Rl are the same
set; both are Boolean rings with the same definition of product, and if +1 denotes the
sum in Rb then we have x +1 y = x(I - y) + (1 - x)y = x + y; hence Rl ~ R, as
required. •
It is clear that Boolean algebra homomorphisms correspond precisely to ring
homomorphisms; we have here an example of a category equivalence: PA and !?4
are mutually inverse functors defining an equivalence between the categories of
Boolean rings and Boolean algebras.
Just as a group can act by permutations on a set, so a ring can act by linear trans-
formations on an additive group. This gives rise to the notion of a module, which
may also be regarded as a generalization of a vector space. Thus let R be any ring.
Bya right R-module one understands an abelian group M, written additively, with
a mapping from M x R into M, (x, a) 1--+ xa, such that for all x, y E M, a, b E R,
(x + y)a = xa + ya, (4.1.8)
x(a+b) =xa+xb, (4.1.9)
, 82 Rings and Modules
x(ab) = (xa)b, (4.1.10)
xl =x. (4.1.11)
Sometimes we have, instead of (4.1.10), the rule x(ab) = (xb)a; in such cases it is
expedient to write the operator on the left, putting ax instead of xa, so this law
now takes the form
(ab)x = a(bx). (4.1.12)
The resulting structure is called a left R-module. Ifwe define the opposite ring of R as
the ring RO whose additive group is the same as that of R but with the multiplication
x.y = yx (x,y E R)
then we can say: a left R-module is a right RO-module and a right R-module is a left
RO -module. In other words, changing the side from which a ring operates on a
module corresponds to a reversal of the order of multiplication in the ring. This
means that instead of using an RO-module we can take the coefficients on the
other side so as to get an R-module. Of course when R is commutative then
RO = R; in this case the difference between left and right R-modules is purely nota-
tional. We shall sometimes write RM resp. MR to indicate that M is a left resp. right
R-module. In general we shall speak of an 'R-module' when the side is not specified,
and a 'module' when we do not wish to specify the ring. A module is called cyclic if it
can be generated by a single element.
As an example we may take the ring itself. Any ring R is a right R-module, with the
action of R being defined by right multiplication. The distributive laws ensure that
(4.1.8), (4.1.9) hold while (4.1.10) follows by the associative law and (4.1.11) is
the property of the unit element. In fact R as right (or left) R-module is cyclic,
generated by 1. Any submodule, i.e. subgroup admitting the right multiplication
by elements of R, is called a right ideal of R. Similarly, left multiplication defines
R as a left R-module, with left ideals as submodules; the ideals defined earlier are
just subsets of R that are both left ideals and right ideals, also called two-sided ideals.
Another example of importance in the sequel is obtained by taking any abelian
group A, written additively, and considering the set E = End(A) of all its endo-
morphisms. If a, bare endomorphisms acting on the right, and x, YEA, then
(4.1.8) just expresses their defining property. We can now define an addition on E
by (4.1.9) and a multiplication by (4.1.10), and can verify that E forms a ring for
these operations, with the identity mapping as unit element. This verification is
straightforward and so may be left to the reader. In this way we obtain a ring struc-
ture on E, called the endomorphism ring of A, and A is seen to be a right E-module.
Given two right R-modules M, N over a ring R, an R-homomorphism f : M -+ N,
also called an R-linear map, is defined as a map f satisfying
f(x + y) = fx + fy, f(xa) = (fx)a for all x, y E M, a E R.
Here we have written f on the left, on the opposite side from the ring coefficients.
We shall usually follow this practice; thus a homomorphism of left modules will
