3.2. An algebraic digression (algebras, rings, ideals)

We assume that the reader is familiar with such notions as group, ring, ideal, field, vector space, isomorphism. A reader not familiar with these notions is advised to consult the textbook [14].
3.2.1. Algebras, rings, ideals
Recall that an ideal in an abstract commutative ring В
is called a subset Г С В, «closed» under the operation of
addition (i.e. а +В Е Г for any а, 6 Е Г) and «absorbing»
multiplication by elements of the ring А, i.e. аб © [ for any а © Г и
ЬЕ В. Here the commutativity property is used'.
An ideal Г С В is called maximal if it is not contained
in any other ideal of the ring В except itself and all of В.
The simplest example: the set of even numbers in the ring of integers
7. This ideal is maximal.
In general, it is very easy to give an example of an ideal in an arbitrary ring В:
it suffices to take any collection of elements е1,...,е, Е В
and let the set Г consist of all combinations of the form
але + :--+а,е,, а1....,@ Е В. (3-1)
Obviously, the set / С В thus constructed is an ideal. Such an
ideal is called finitely generated, and the elements е1,....е, its generators. Notation: 1 = (е1,...,е,). An ideal generated by one
'The commutativity of the ring shows up in the fact that in the condition аб Е Г the order
of multiplying the elements а Е Ги фЕ Л does not matter at all. In noncommutative
rings there are left, right, and two-sided ideals.
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element is called principal. For example, the set of even numbers
in the ring 2 forms a principal ideal generated by a single generator - the number 2. Of course, there are also non-finitely-generated
ideals.
Obviously, any finitely generated ideal Г can be specified using various systems of generators. A set of generators е1,...,е,
we shall call minimal if none of its elements can be discarded, i.e. discarding any element, the remaining ones already
do not generate the ideal Г. At first glance, the notion of a minimal
set of generators looks very much like a basis of a vector space,
however there is an essential difference between them. For example, there exist ideals in which the minimal sets contain different
numbers of generators (see the following example).
EXAMPLE 3.2. Let В = Вт,у| be the ring of polynomials in two
variables. Consider in it the ideal Г = (е1,е2,ез), generated by
three generators:
е =зу, е› = (тф+у- Пу, ез= (т-уу. (3.2)
First we show that Г = (у), i.e. the ideal Г can be specified by means of
a single generator у. Since all е; are divisible by у, any element
а Е Г is also divisible by у, hence the inclusion ГС (у) holds.
To prove the reverse inclusion, it suffices to show that у Е Г.
This is done by a simple check: у = 2е; — е› — ез. Thus we
have proved that the ideal [= (у).
At the same time, the set of three generators (3.2) cannot be reduced to one. Indeed, for no 1 does the ideal (е;) contain the monomial у,
and consequently Г = (е;). Moreover, one can show that the set of generators (3.2) cannot be reduced even to two (the reader is left to check this as an exercise).
Let А — be a commutative algebra, i.e. simultaneously a commutative ring and a vector space (over some field К),
where the operations of multiplication by elements of the field К and of the ring В = А
are consistent:
(Ка)ь = а(К5) = К(а5) УЕЕК, УМа,5Е В.
The dimension of the algebra А is the dimension of А as a vector
space.
In a finite-dimensional algebra of dimension п there is a basis е1,..., ев
(a basis of the corresponding vector space), and the product
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of any two elements of the algebra can be defined via the products of its basis elements:
т
— [ ги ее; = > Ау е1, 1 =1,..., п.
[=1
The set of numbers А, „.Г=1,...,П, is called the «multiplication table»
of this algebra in the given basis and uniquely determines the products
of any of its elements:
ть т т
а = › Озе:, р = › В;е; —> аб = у. оз Вузе.
1=1 1=1 1,1.1=1
An algebra А (of finite or infinite dimension) is called
graded if a decomposition of the vector space А is given
as a direct sum of its subspaces
А=АфФА: Ф...ФА,, АССА, (3-3)
such that the following condition holds: for any elements а Е А; and фЕ А; their
product аб Е А; ,;, if + 7 < К, and аб = 0, if + Е > К.
In the case of an infinite-dimensional algebra the direct sum (3.3) may also be
infinite: = co. Such a decomposition is called a grading
of the algebra А.
EXAMPLE 3.3. The simplest example of a graded algebra is the algebra В [т1,...,т»„| of polynomials in п variables over the field of real numbers, or, more generally, the algebra К] т1,...,т| of polynomials in п variables over an arbitrary field К. Obviously, for such an algebra
the grading (3.3) holds with Ё = co, where the subspace А; consists of homogeneous polynomials of degree 1, also called forms
of degree 1, or, for short, 1-forms. We shall meet another example of an algebra admitting a finite grading below.
An ideal of the algebra А is an ideal Г of the ring А that is at the same time a subspace of the vector space А.
EXAMPLE 3.4. Let А be the algebra of germs of smooth functions of
the variables т1,..., 1, and let the ideal ГС А consist of the germs of functions
that vanish at the point 0. This ideal is obviously finitely generated. As its generators one can take
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the germs of functions е1 = 11,...,е, = т,. Indeed, by Hadamard's lemma, every germ РЕ Г has the form (3.1) with some а1,..., а, from
the ring А. This proves that Г is an ideal in the ring А, but it is also easy to check that Г is a subspace of the vector space
А. Thus Г is an ideal in the algebra А.
Note that the set of generators е; = т; is not a basis of
Г as a vector subspace. Indeed, the vector subspace У С А, generated by linear combinations of the vectors
е; = т; consists only of germs of linear functions, whereas
Т obviously also contains nonlinear ones. One can also compare the dimensions of У and Г as vector spaces: 4па У = г, but Ап Г = 55.
PROBLEM 3.1. Find other systems of generators of the ideal
ГС А from Example 3.4.
3.2.2. Factorization (the general idea)
DEFINITION 3.2. An equivalence relation is given on a set А if for each pair of elements т and у of this set,
one of two things is defined: either the elements хи / are equivalent (denoted 1 ^> у), or they are not equivalent (1 7 у), and moreover the following properties must
hold: reflexivity (5 -> <), symmetry (5 ^- у), transitivity (if 5 м уиу-> 5, then д-р).
If such a relation is given, then the set А can be represented as a
union (finite or infinite) of subsets Ао, called equivalence classes and having the following properties:
1) each element 1 Е А is contained in some subset
А. С А, and only in one. In other words, the subsets А»
do not intersect, and their union (finite or infinite) gives
the whole set А;
2) the elements т and у are equivalent (7 ^> у) if and only if they
belong to the same subset Ао.
Obviously, conversely, if a representation of the set А
as a union of subsets Ао satisfying the first property is given, then, setting т ^> у if and only if these elements belong to the same subset А, we thereby obtain
a certain equivalence relation on А. The sets А’ are called equivalence classes; they can also be regarded as
elements of some set, which is denoted А/ -> and is called
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the quotient set of the set А with respect to the given equivalence
relation.
The definition given above is completely abstract. It acquires concrete content when we specify a way of defining the equivalence. We already encountered one meaningful example of factorization of a set above, when we defined germs of functions.
In singularity theory of smooth mappings, the objects of study are special types of equivalence, which will be introduced in section 8. Let us give one more example, of a purely geometric nature.
EXAMPLE 3.5. If we take as А the set of nonzero vectors of three-dimensional space (all vectors are assumed to emanate
from the origin 0) and regard as equivalent vectors lying
on the same ray, the elements of the quotient set А/ -> will be
all possible rays emanating from the point 0. In order to better
picture this set, consider the sphere of unit radius centered at 0. Each ray intersects it at exactly one
point, so the set А/- can be identified with the points of the sphere. This map is not only one-to-one, but
also continuous together with its inverse, i.e. a homeomorphism. Hence the quotient set А/ -- is homeomorphic to the ordinary two-dimensional sphere.
If for the same set А we regard collinear vectors as equivalent, we arrive at a new and very substantive geometric object. Obviously, the elements of the quotient set
А/ -> will be all possible lines passing through the point 0. Such a
set is called the projective plane, and the lines composing it
are called points of the projective plane. The points of the projective plane can no longer be identified with the points of the sphere, since each
line intersects the sphere twice. One can say that the projective
plane is a sphere with opposite points «glued» together,
but this is unlikely to help one understand what it looks like.
REMARK 3.1. There is a geometric construction that allows one to represent the projective plane more visually, as
a Möbius band with a disk glued to it according to the «boundary to
boundary» principle?. This shows that the quotient sets А / --, constructed
by means of the two equivalence relations listed, despite their outward similarity, differ very strongly. For example, the sphere is orientable, while the projective plane, like the Möbius band, is not
(or, as is often said, «has only one side»).
?See, for example, the textbook [44], ch. [Х, sec. 9.4.
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3.2.3. Quotient algebras
Of particular interest is the case where А is not merely a set, but some algebraic structure: a group, a ring, an algebra,
etc. Then it is natural to try to define the equivalence relation in such a way that the quotient set А/-> is an algebraic structure of the same type as А.
Let 4 = А be a commutative ring and ГС В an ideal in it.
If in В we consider only the operation of addition (forgetting about
multiplication), then В is a group, and ГС В its additive subgroup. The coset of an element а of the group В with respect to the subgroup /[ is the set
а-+1= {а+6 | УБЬЕТ.
Obviously, for two elements а and а’ the cosets аЁГиа’ +1
either coincide (if а—а' Е Г), or do not intersect (if а-а' @ Г).
Thus we obtain an equivalence relation: @ г а’, if
and only if а-а’ Е Г.
The quotient group of В by Г (denoted В/Г) is the quotient set
В/ -> with respect to this equivalence relation, consisting
of all cosets of the group В with respect to the subgroup Г, on which the operation of addition is defined by the formula (а + Г) + (а' + Г) = (а-+ а’) + Г.
It is easy to check that the set Д/Т with the addition operation defined in this way is an additive group (in particular,
the zero element is Г).
Now recall that in the ring В besides addition there is also multiplication, and these operations are related by the distributivity property. This allows us to define multiplication also in the quotient group В/Т by the formula
(а-+ (а +1 =аа' + Г.
PROBLEM 3.2. Prove that the operation of multiplication of cosets defined in this way is distributive with respect to addition, if ГС В is an ideal (the mere condition that Г is a
subring is not sufficient for this).
Thus, in the case when Г is an ideal (which will always
be assumed from now on), the quotient set В/ [ with the addition and multiplication operations defined above is a ring. It is called the
quotient ring of the ring В by the ideal Г.
EXAMPLE 3.6. Consider the quotient set &/Т, where ГС #, is the ideal
consisting of the even numbers. It is easy to see that 2/1 contains
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exactly two equivalence classes, consisting of the even and odd
numbers, which can be denoted 0 and 1 respectively. It is also easy
to check that the addition and multiplication of these elements are performed
according to rules that make the set {0,1} the residue field 2. Thus, 2/Т is the residue ring 2, and since the latter is not
only a ring but also a field (every nonzero element is invertible), the
ideal ГС # is maximal.
Now let / С А be an ideal of the algebra А, i.e. an ideal of the ring А and
at the same time a subspace of the vector space А. In this
case one can turn the quotient ring А/Г into a quotient algebra. To do
this one needs to make the ring А/7 a vector space (over the field К). Addition of vectors in А/7 is already defined (as addition of elements of the ring), and it remains only to define the operation of multiplication
of elements of А/Т by elements of the field К in such a way that all the axioms of a vector space hold. This is done by the rule
Кк(а+Г =кКа+Т, УаЕА, КЕК.
Note that here we use the fact that КБЕ Г for any ВЕ Ги
КЕ К, since ГС А is a vector subspace. It is easy to check (this is left
to the reader as an exercise) that all the axioms of a vector space hold and, moreover,
the operations of multiplication by elements of the field К and of the ring are consistent. The algebra obtained in this way is called the quotient algebra А/Г.
EXAMPLE 3.7. Let А be the algebra of germs of smooth functions (or
polynomials) in п variables, and let the ideal Г С А consist of the germs
of functions (respectively, polynomials) that vanish
at the point 0. Then the quotient set А/7 consists of the cosets
+ Г ГЕА, each of which is uniquely characterized by the value /(0), i.e. by a single real number. It is easy to see that
the correspondence ({} + Г) =} (0) defines a map 4/7 — В, which is an isomorphism (of rings, of vector spaces, and hence,
of algebras). Thus А// is a quotient algebra, isomorphic
to the algebra ЕВ.
EXAMPLE 3.8. Let А be the algebra of germs of smooth functions (or
polynomials) in one variable, and let the ideal ГС А consist of the germs
of functions (respectively, polynomials) /(1), satisfying the condition /(0) = /"(0) = 0. From Hadamard's lemma it follows that any such
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function can be represented in the form /(т) = 229(х), 9 Е А, and, consequently, the
ideal [ is generated by a single element - the monomial #2, i.e. А/Г = А/(2?).
The quotient set А/(=?) consists of the cosets {+ Г, Е А,
each of which is uniquely characterized by the values {(0} and
|’ (0), i.e. by two real numbers. Assigning to each
coset / + Г the pair of numbers #(0), {'(0) defines a map
А/(л?) + В?, which is obviously an isomorphism of А/(1?) and
Е? as vector spaces, but does not define an isomorphism of А/(1?)
as an algebra. In order to turn А/(5?) into a quotient algebra,
one needs to define correctly the operation of multiplication in it.
For definiteness let us first consider the case of polynomials. Let us turn Е[2|/(22) into the so-called algebra of truncated polynomials
below second degree, which, as the name suggests, consists of
polynomials of degrees 0 and 1. Addition and multiplication by numbers of the elements of this algebra are defined in the usual way. The product of the polynomials а1 + 1х and а2 + ох equals ала2 + (а165 + а261)т, in other words,
it is obtained from the ordinary product of polynomials by discarding all monomials of degree higher than 1, i.e. it is the remainder of the ordinary product of the polynomials upon division by 22. In the algebra Е[| / (22)
there is a natural grading: В|1|/(22) = АоФ А|, where А; is the subspace of homogeneous polynomials of degree #.
Now let А — be the algebra of germs of smooth functions of one variable. It is easy to check that the map А/(2?) -+ В |/ (22),
which assigns to each coset / + Г the polynomial
1(0)- /'(0)х, is not only an isomorphism of А/(1?) and В[1|/ (22) as
vector spaces, but also defines in А/(12) a multiplication turning the quotient set А/(1?) into an algebra, and thus is an isomorphism of А/(2?) and В[1|/ (17) as algebras.
PROBLEM 3.3. Write out the multiplication table in the algebra of truncated
polynomials below second degree, taking as a basis the monomials
е1 =Ти е2 =2.
Answer. If instead of the coefficients АН of the expansion of the product е;е; in the basis е1,е2 we write in the cells the products themselves,
we obtain the following table:
1х
111 т
|0
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REMARK 3.2. The algebra of truncated polynomials below second
degree is sometimes also called the «algebra of functions on a doubled point». This name is explained as follows. Consider the set of functions defined on the «doubled point» — a set containing only two points, for example, +=. Obviously, this
set is an algebra (with the standard operations of addition and multiplication: the values of the
function at each of the two points are added and multiplied) of dimension 2.
As a basis of this algebra one can choose, for example, the so-called delta-functions, equal to 1 at one point and zero at the other (the reader is invited, as an exercise, to write out the multiplication table
in this basis). But one can also choose the basis е1 = 1, е› = 5. These functions in
our case are the restrictions of the corresponding monomials
from the whole real line to the doubled point +. As a simple
exercise the reader is invited to write out the multiplication table in
this basis and to verify that in the limit = -+ 0 it coincides with the multiplication table given above in the algebra В / (22).
Similarly, for any integer & > 2 one defines
Е[=]/(х^)) - the algebra of truncated polynomials below degree &.
PROBLEM 3.4. Describe the algebra В |/(х^), its multiplication table
in some basis, and its grading. Show that В /(х^) is isomorphic to the quotient algebra А/Т, where А is the algebra of germs of smooth functions of
one variable, and the ideal ГС А consists of the germs {(5), satisfying the condition (0) = {'(0) =... = #70 (0) = 0.
REMARK 3.3. The construction described above gives a universal way to turn any finite-dimensional vector space over an arbitrary field К into an algebra.
Let У be a vector space of dimension К; it is isomorphic to
the space А, consisting of the polynomials
Р(Е) = ао + аа +... ава", м ЕК.
Having chosen bases in \ and А, one can construct explicitly an isomorphism ф: У — А. For example, let us assign to the vector е; of an arbitrary basis in У
the element #1" of the space А: ф(е;) =#`! for =1,...,К. Using the
procedure described above, А is turned into the algebra of truncated polynomials К] 2] /(х^). After this it remains to define the multiplication of two
elements ‚© of У by the formula (и) = ф(и)ф(о).