Abstract Algebra 17
RingS of ENDOMORPHISMS
AND QUATERNIONS
Objectives
From this unit a learner is expected to achieve the following
1. Recall the definitions of homomorphism and isomorphism between groups.
2. Learn the definition of endomorphism.
3. Study that the set of endomorphisms of an abelian group forms a ring.
4. Familiarises skew field with an example
Sections
1. Introduction
2. Definitions
3. Rings of Endomorphisms
4. Example of a skew field: The Quaternions of Hamilton
Introduction
An endomorphism of an an abelian group A is a homomorphism from A into itself. In this session we will establish that the set of endomorphisms of an abelian group forms a ring. We also discuss that the quaternions form a skew field under addition and multiplication. Also the statament of Wedderburn’s Theorem is discussed.
Definitions
Let be groups. A mapping is called a homomorphism if
.
If is one-to-one (i.e., injective) then it is an isomorphism (or monomorphism) of into If is onto (i.e., surjective) then it is an epimorphism of onto
If is both one-to-one and onto (i.e., bijective) then it called an isomorphism of onto in that we say that isomorphic to and is denoted by
A homomorphism of an abelian group into itself is called an endomorphism of . An endomorphism of that is both one-to-one and onto is called an automorphism of .
Rings of Endomorphisms
Let A be an additive abelian group. We denote the set of all endomorphisms of A by Hom(A) or by . For Hom (A), the homomorphism addition is defined in terms of its values on each i.e., by ;
and the homomorphism multiplication (i.e., function composition) is defined by
We now establish that set of endomorphisms of an abelian group forms a ring.
Theorem 1 Let A be an additive abelian group. The set Hom (A) of all endomorphisms of A forms a ring under homomorphism addition and homomorphism multiplication.
Proof
We first show that Hom (A) is an abelian group under homomorphism addition.
- First of all we show that is a homomorphism.
,
by the definition of homomorphism addition
, since and are homomorphisms
, since A is an abelian group.
, again by the definition of homomorphism addition
Hence is a group homomorphism from A to A and hence Hom (A). Thus closure property of the operation “homomorphism addition” in the set Hom (A) is verified.
, by the commutativity in A
Hence
and the addition in Hom (A) is commutative.
- ForHom(A),we have
- If e is the additive identity of A, then the homomorphism 0 defined by
for is an additive identity in Hom (A). This can be seen as follows:
For Hom (A) , and we have
Since Hom (A) , and are arbitrary elements, the above shows that
Hom (A) , and we have for Hom (A).
- For Hom (A), defined by is in Hom (A), since ,
and Thus is the additive inverse of in Hom (A).
Combining the above, we have < Hom (A), + > is an abelian group.
The verification that Hom (A) is closed under homomorphism multiplication is left as an assignment.
We now verify that the distributive laws of homomorphism addition and homomorphism multiplication hold in Hom (A).
since is a group homomorphism.
by the definition of homomorphism multiplication.
Thus
Also, the right distributive law follows from the following:
This completes the proof of the theorem.
The following example illustrates that the above ring Hom (A) need not be commutative.
Example 1 Consider the abelian group . We can specify an endomorphism of this group by giving its values on the generators (1, 0) and (0, 1) of the group . Define
by and .
Define
by and .
Note that maps every thing onto the first factor of and collapses the first factor. Thus
while
Hence
Example 2 Let F be a field, and let be the additive abelian group of the ring , the ring of polynomials with coefficients in F. In this example, to simplify the notation, we denote the additive group by . We consider
· One element of acts on each polynomial in by multiplying it by x. Let this endomorphism be X, so
Verifying X is an endomorphism is left as an assignment.
· Another element of is formal differentiation with respect to x. The familiar formula “the derivative of a sum is the sum of the derivatives” guarantees that differentiation is an endomorphism of . Let Y be this endomorphism, so
· It can be seen that (the unity in) so that where X and Y as above.
· Multiplication of polynomials in by any element in F also gives an element of .
Theorem 1 If is any multiplicative group, and R be any commutative ring with unity and
then is a ring.
Proof
- We define the sum of two elements in by
Observe that except for a finite number of indices i, so is an element of . It can be easily verified that is an abelian group with additive identity
- Multiplication of two elements of is defined by the use of the multiplication in G and R and is as follows:
Again at most a finite number of are nonzero. Thus multiplication is closed on
- The distributive laws follow at once from the definition of addition and the formal way we used distributively to define multiplication. For the assocaitivity of multiplication
This completes the proof of the theorem.
Definition (Group Ring) The ring defined above is the group ring of G over R.
Definition (Group Algebra) If F is a field, then is the group algebra of G over F.
Example 3 The group is cyclic of order 2 (with generator a). Since is a field, is the group algebra and is given by
If we denote the elements in , in the obvious natural way, by
,
the addition and multiplication tables are given below:
In the above, note that
Example of a skew field: The Quaternions of Hamilton
Definition (Skew field) A noncommutative division ring is a skew field.
We consider , the field of real numbers under addition. We take
, the direct product of .
We define addition on by addition by components. It can be verified that under this addition by components is an abelian group.
Let us rename certain elements of Q. We shall let
and
We furthermore agree to let
and
In view of our definition of addition, we then have
Thus
To define multiplication on Q, we start by defining
and
(Note the similarity with the so-called cross product of vectors. These formulas are easy to remember if we think of the cyclic sequence )
We define a product to be what it must be to make the distributive laws hold, namely,
Theorem 2 The quaternions Q form a skew field under addition and multiplication.
Proof The proof includes the discussion just above and the following. Since we see that hence multiplication is not commutative, so Q is definitely not a field. The only axiom that we prove is the existence of a multiplicative inverse for with not all Computation shows that
If we let and we see that
is a multiplicative inverse for a. The verification of other axioms are left as an assignment.
We conclude this session by stating the famous Wedderburn’s Theorem.
Theorem 3: Wedderburn’s Theorem A finite division ring is a field.
Remark: In view of Wedderburn’s theorem we conclude that there are no finite skew fields.
5. Summary
In this session we have seen that the set of endomorphisms of an abelian group forms a ring. We have described that the quaternions form a skew field under addition and multiplication. Also the statement of Wedderburn’s Theorem have been discussed.
Assignments
1. Find an example to show that a finite multiplication subgroup of a strictly skew field need not be cyclic.
2. Show that the ring is not isomorphic to the ring
3. Prove Wedderburn’s theorem.
Quiz
1. If is isomorphic to ______
(a).
(b).
(c).
(d). None of the above
Ans. (a)
2. Which of the following is a field?
(a). The Quaternions of Hamilton
(b). where p is a prime
(c).
(d).
Ans. (b)
3. Which of the following is always cyclic?
(a). A finite multiplicative subgroup
of a strictly skew field.
(b). A multiplicative subgroup of a field.
(c). A multiplicative subgroup of
an integral domain.
(d). None of these.
Ans. (b)
FAQ
1. Given an abelian group. Is there any ring associated with it?
Ans. Yes. The set Hom (A) of all endomorphisms of A forms a ring under homomorphism addition and homomorphism multiplication.
2. Is there any ring that is not commutative?
Ans. Yes. For example, the ring of all matrices with entries in a field is not commutative.
3. Is there any ring End (A) that is commutative?
Ans. Yes. For example, the ring is commutative.
4. Is there any division ring that is not commutative?
Is there any division ring that is not commutative?
Ans. Yes. The quaternions of Hamilton gives a non-commutative division ring. It is a strictly skew field.
Glossary
Group Homomorphism: Let be groups. A mapping is called a homomorphism if
.
Group Monomorphism: If is one-to-one (i.e., injective) then it is an isomorphism (or monomorphism) of into If is onto (i.e., surjective) then it is an epimorphism of onto
Group Onto Isomorphism: If is both one-to-one and onto (i.e., bijective) then it called an isomorphism of onto
Isomorphic Groups: If there is an isomorphism from G onto H, we say that is isomorphic to and is denoted by
Endomorphism: A homomorphism of an abelian group into itself is called an endomorphism of .
Automorphism: An endomorphism of that is both one-to-one and onto is called an automorphism of .
Rings of Endomorphisms: Let A be an additive abelian group. The set of all endomorphisms of A, denoted by Hom(A) or by , is a ring under the following operations and is called a ring of endomorphisms. For Hom (A), the homomorphism addition is defined in terms of its values on each i.e., by
;
and the homomorphism multiplication (i.e., function composition) is defined by
Group Ring: The ring defined above is the group ring of G over R.
Group Algebra: If F is a field, then is the group algebra of G over F.
Skew Field A noncommutative division ring is a skew field.
REFERENCES
Books
1 I. N. Herstein, Topics in Algebra, Wiley Eastern Ltd., New Delhi 1975.
2 Nathan Jacobson, Basic Algebra Vol 1, Hindustan Publishing Corporation (India), Delhi 1993.
3 John B. Fraleigh, A First Course in Abstract Algebra, Fifth Edition, Addison-Wesley, California, 1999.
4 Vijay K. Khanna, S. K. Bhambri, A Course in Abstract Algebra, Vikas Publishing House Pvt. Ltd., New Delhi, 1996.
5 N. S. Gopalakrishnan, University Algebra, Wiley Eastern Limited, New Age International Limited, New Delhi 1995.
6 Surjeet Singh, Qazi Zameeruddin, Modern Algebra, Vikas Publishing House Pvt. Ltd., New Delhi 1997.
7 Shanti Narayan, A Text Book of Modern Abstract Algebra, S. Chand & Co., New Delhi.
8 Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House, 1997.
10/ 8.7.2012