Download MATH20212: Algebraic Structures 2

MATH20212: Algebraic Structures 2
This course follows on from Algebraic Structures 1 MATH20201, so some
basic knowledge and understanding of groups is assumed. This course concentrates on the algebraic structures called Rings and Fields.
The central aims of the course are:
• that you become acquainted with the basic examples and the general ideas,
to the point where you have internalised the main concepts and can use them
in new situations;
• to introduce the general notion of a homomorphism (a structure-preserving
map) of rings and its kernel;
• to introduce the factor, or quotient, construction.
Like Algebraic Structures 1, there is a great deal of emphasis on examples:
subrings of Z, Q and C; congruence class rings; polynomial rings; matrix
rings. Polynomials are the central examples. Algebraic Structures 1 and 2
together give a foundation for further courses in algebra, as well as those
(e.g. some in geometry and topology) which use algebraic structures.
These notes cover the material of the whole course. Lectures will be
used for explaining the concepts and giving further details of the proofs and
examples. There will be example sheets to accompany the notes. Work on
the examples relevant to what has been covered recently in lectures: don’t
get behind (you can always come back and use those you skipped over for
revision). The tutorials will be used for going over extra examples and should
be regarded as being as important as the lectures. Prepare for the class (try
the example sheets beforehand) and be ready to participate (asking questions,
suggesting solutions).
Further reading is not necessary but might be interesting and/or useful.
There is the course text by Fraleigh. See the course website for further
details.
Finally, please let me know of any errors/typos you find in the notes,
examples, solutions.
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Rings - Definitions and Examples
Basic idea: a ring is an “arithmetic system” with two binary operations,
usually written + and ×. The integers are a ring with these operations. The
ring of integers modulo n is another example, as is the ring of n × n square
matrices with integer entries. There are many more examples.
Definition 1.1. A ring is given by the following data: a set R and two
binary operations, written + and ×, on R. The conditions to be satisfied are:
(R1) < R, + > is an abelian group with identity 0.
(R2) × is associative.
(R3) × is distributive over +:
ie. a × (b + c) = (a × b) + (a × c) and (b + c) × a = (b × a) + (c × a)
for all a, b, c ∈ R.
(R4) there exists an element 1 ∈ R, different from 0, that is an identity for
×.
Notes:
• We write < R; +, ×, 0, 1 > if we want to show all the data but usually
we don’t want to do that, so write just R.
• We usually write ab instead of a × b.
• Note that + is always commutative in a ring but × may not be commutative. If × is commutative we call R a commutative ring.
• In some textbooks rings are not assumed to have a multiplicative identity. However in this course “ring” will mean “ring with a 1”.
Examples 1.2.
(1) Z, the ’mother of all rings’.
(2) Zn = {0, 1, ..., n − 1}, the ring of integers modulo n with addition and
multiplication modulo n . More formally we think of this ring as the
set of equivalence classes {[0]n , [1]n ..., [n − 1]n } where
[a]n = {b ∈ Z | b ≡ a mod n}.
We know that < Zn , + > is an abelian group from Algebraic Structures
1 (in fact it is cyclic) and we know from first year that multiplication
modulo n is associative and distributive over addition modulo n.
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(3) Mn (R), the ring of all n × n matrices with entries in R, with addition
and multiplication of matrices. From Linear Algebra we know that
the axioms (R1)-(R4) hold. Note that this is our first example of a
non-commutative ring, if n ≥ 2.
Other matrix rings include Mn (Z), Mn (Q), Mn (C) and Mn (Zm ).
(4) An unusual example: for any set X 6= ∅ the power set P(X) is a ring
with addition and multiplication defined by
A + B = (A ∪ B)\(A ∩ B),
AB = A ∩ B.
Here 0 = ∅, 1 = X and −A = A for all A ⊆ X.
Our first result is completely obvious in most examples of rings. The
point is to show that its three statements hold in the most general situation
as consequences of the ring axioms (R1)-(R4).
Lemma 1.3. Let R be a ring. Then, for all a, b ∈ R
(i) 0a = a0 = 0
(ii) a(−b) = (−a)b = −(ab)
(iii) (−a)(−b) = ab.
Proof. (i) We have
0a + 0a = (0 + 0)a = 0a
and so be adding −0a to both sides we get 0a = 0. Likewise a0 = 0.
(ii) By definition −(ab) is the unique element in R with the property that
ab + (−(ab)) = 0. Now
ab + (−a)b = (a + (−a))b = 0b = 0.
Hence (−a)b = −(ab) by uniqueness. Similarly a(−b) = −(ab).
(iii) By using (ii) twice and −(−a) = a we get
(−a)(−b) = −(−(ab)) = ab.
Some notation:
• We usually write a − b instead of a + (−b).
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• For n ∈ N and a ∈ R we define
n·a=a
|+a+
{z... + a}
n
and we can extend this definition to arbitrary integers by using 0·a = 0
and setting n · a = (−n) · (−a) if n < 0. With this notation we have
the following rules for multiples:
(n + m) · a = n · a + m · a
n · (a + b) = n · a + n · b
n · (m · a) = (nm) · a
where n, m ∈ Z and a, b ∈ R.
Definition 1.4. Let R be a ring and S ⊆ R. Then S is a subring of R if it
is a ring in its own right with respect to the same addition and multiplication
as in R.
By the lemma below, to show a subset is a subring, it is enough to check
that it contains 0 and 1 and is closed under taking negatives and under the
operations, + and ×.
Lemma 1.5. If R is a ring and S ⊆ R is a subset then S is a subring of
R, iff:
• 1 ∈ S;
• r, s ∈ S implies r + s, r × s ∈ S;
• r ∈ S implies −r ∈ S.
Proof. If S is a ring then the three conditions are satisfied by definition.
Suppose S satisfies the three conditions of the lemma. Let r, s ∈ S. Using
the subgroup test, < S, + > is a subgroup of < R, + > by the second and
third conditions.
Commutativity of +, associativity of × and distributivity of × over +
are inherited from R.
Since 1 ∈ S, then −1 ∈ S by the 3rd condition and then 0 = 1+(−1) ∈ S
by the 2nd condition. Since 1 6= 0 in R, we have 1 6= 0 in S. Therefore S
satisfies (R1)-(R4) and is a ring.
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That was loosely said: what we should have said was the following. “If
(R; +, ×, 0, 1) is a ring and S ⊆ R then (S; +0 , ×0 , 0, 1) is a ring, where +0 is
the restriction of the operation + to S and ×0 is the restriction of × to S, iff
(list of conditions as in the lemma)” or, more briefly, “If (R; +, ×, 0, 1) is a
ring and S ⊆ R then S, equipped with the operations inherited from R, is a
ring iff (list of conditions as in the lemma)”.
Anyway, the point is, to check that a subset of a ring S is a subring you
don’t have to check everything: many of the conditions come for free because
they hold everywhere in the ring S.
Example 1.6. Z[i] = {a + bi | a, b ∈ Z}, where i denotes a square root of
−1, is a subring of C.
Examples 1.7. More examples of rings: Small rings can be presented by
giving their “addition and multiplication tables”. Here are two examples.
Can you identify the rings?
+
0
1
2
3
0
0
1
2
3
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
0
1
+
0
0
1
1
1
0
x
x
1+x
1+x 1+x
x
×
0
1
2
3
0
0
0
0
0
x
x
1+x
0
1
1
0
1
2
3
2
0
2
0
2
1+x
1+x
x
1
0
3
0
3
2
1
×
0
1
x
1+x
0
0
0
0
0
1
x
0
0
1
x
x
1+x
1+x
1
1+x
0
1+x
1
x
There is an issue here: these tables certainly specify two operations on
each relevant set and it would be easy to pick out the 0 and 1 even if they
had not been identified in name, but it could be rather tedious to check that
all the conditions (associativity and distributivity in particular) for a ring
are satisfied. Of course you could give the tables to a computer to check. We
will come back to these examples and, by recognising what they are, we will
be able to avoid the tedious case-by-case checking of the ring axioms.
Definition 1.8. Polynomial Rings
Let R be a ring. The ring of polynomials R[X] in the indeterminate
X is defined as follows:
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Elements: Formal linear combinations of the form
X
ai X i
i≥0
where the coefficients ai ∈ R for i = 0, 1, ... and only finitely many of these
coefficients are non-zero. When writing specific polynomials we normally
don’t include the terms with zero coefficients.
Equality:
X
X
ai X i =
bi X i ⇔ ai = bi for all i ≥ 0.
i≥0
i≥0
Addition:
X
ai X i +
X
i≥0
bi X i =
i≥0
X
(ai + bi )X i .
i≥0
Multiplication:
!
X
!
X
ai X i
i≥0
bi X i
=
i≥0
The zero element is
P
i≥0
!
X
X
k≥0
i+j=k
ai b j
X k.
0X i = 0 and the one is 1X 0 +
P
i≥1
0X i = 1.
Example 1.9. Let R = Z4 . Then 2 + X + 2X 2 , 3X + 2X 2 ∈ Z4 [X] and
(2 + X + 2X 2 ) + (3X + 2X 2 ) = 2 + 4X + 4X 2 = 2
(2 + X + 2X 2 )(3X + 2X 2 ) = 6X + 4X 2 + 3X 2 + 2X 3 + 6X 3 + 4X 4
= 6X + 7X 2 + 8X 3 + 4X 4 = 2X + 3X 2 .
An important property of a polynomial is its degree. For f =
we define
P
i≥0
ai X i
deg(f ) = the largest i such that ai 6= 0 and we let deg(f ) = −∞ if f = 0.
For any f, g ∈ R[X] we have
deg(f + g) ≤ max(deg(f ), deg(g))
deg(f g) ≤ deg(f ) + deg(g).
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Indeed if deg(f ) = m and deg(g) = n, then
f=
m
X
i
ai X ,
g=
i=0
n
X
bi X i
i=0
where am 6= 0, bn 6= 0 and
f g = a0 b0 + (a0 b1 + a1 b0 )X + ... + am bn X n+m .
So X n+m will be the highest power of X with a non-zero coefficient unless
am bn = 0. This happened in the example with Z4 [X] above.
More generally we can define polynomial rings in more than one variable.
For n > 1 we define
R[X1 , ..., Xn ] = R[X1 , ..., Xn−1 ][Xn ].
Hence the elements of R[X1 , ..., Xn ] are polynomials in Xn with coefficients
in R[X1 , ..., Xn−1 ].
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