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EXERCISE 1.2
1. Write down a pair of integers whose:
(a) sum is - 7 (b) difference is -10 (c) sum is 0
2. (a) Write a pair of negative integers whose difference gives 8.
(b) Write a negative integer and a positive integer whose sum is -5.
(c) Write a negative integer and a positive integer whose difference is -3.
3. In a quiz, team A scored - 40, 10,0 and team B scored 10,0,- 40 in three successive
rounds. Which team scored more? Can we say that we can add integers in
any order?
make the following statements true:
Answers
MA501
Fall 2006
Algebra for Teachers
1 Algebra
1.1 Groups, Rings, and Fields
Many of the quantities we deal with in algebra satisfy similar properties and so it is convenient to set up a naming system to keep them straight. The most basic algebraic systems we will deal with are groups, sets on which there is a single operation. More formally,
Definition 1.1.1: (Abelian Group) An Abelian group is a set G together with a binary operation defined on G which satisfies a certain number of properties. These properties will be expressed in terms of the operation +:
Addition is commutative: a+b=b+a for all a in G .
Addition is associative: (a+b)+c=a+(b+c) for all a , b , c in G .
G contains an additive identity 0, i.e. there is an element 0 in G such that a+0=a for all a in G .
Every element of G has an additive inverse, i.e. for every a in G , there is a b such that a+b=0 . An additive inverse of a is often denoted -a .
Example 1.1.1: Some examples of Abelian groups are the set of ordinary integers under addition and the set of non-zero rational numbers under multiplication. If n is a positive integer, then the set {0,1,..,n-1} can be made into an Abelian group by defining a+b to be the remainder when you divide sum of the integers a and b by n . This group is called the group of integers modulo n . You can also define a multiplication a·b as the remainder when the product of the integers a and b is divided by n ; the set {1,2,…,n-1} is an Abelian group if and only if n is a prime number.
Exercise 1.1.1: Show that if n is not prime, then the non-zero integers modulo n do not form a group under multiplication.
Exercise 1.1.2: Fill in the details of the following sketch of the converse. Suppose n is a prime and r is in G={1,2,…,n-1} . One can define a map f from G to itself by f(g)= the remainder when r·g is divided by n . This is one-to-one because if f(g)=f(h) , then r·g-r·h=r·(g-h) is divisible by n . Since n is prime and r is not divisible by n , it follows that g=h . But then the map f is also onto and so there is a g in G with f(g)=1 .
Remark 1.1.1: Although the above proof shows that there exists a multiplicative inverse, it gives no indication of how to go about finding one. Of course, there are only finitely many possibilities, and so one could just try each number from 1 to n-1… . This situation will be fixed in the next section.
Proposition 1.1.1: An Abelian group has exactly one additive identity.
Proof: The definition says that every Abelian group has an additive identity. Suppose that 01 and 02 are identities of some Abelian group. Then 01+02=01 because 02 is an additive identity and similarly 02+01=02 since 01 is an additive identity. But commutativity implies that 01+02=02+01 . So, combining results one obtains
01=01+02=02+01=02
which shows the result.
Proposition 1.1.2: Every element a of an Abelian group has exactly one additive inverse.
Proof: The definition guarantees that every element a has at least one additive inverse. Suppose that b and c are both additive inverses of a . Then, one has:
b=b+0=b+(a+c)=(b+a)+c=(a+b)+c=0+c=c+0=c
Each step can be justified by one of the following reasons: (i) b and c are additive inverses of a , (ii) addition is commutative, or addition is associative. Make sure you can label each equal sign with the reason it is true.
Since every element a of an Abelian group G has a unique additive inverse -a , one can define the subtraction operation by a-b=a+(-b) .
Rings are sets with two operations, usually denoted + and · :
Definition 1.1.2: (Commutative ring with identity) A commutative ring with identity is a set R with two binary operations + and · which satisfy:
R and Addition + is an Abelian group.
R and Multiplication · is commutative and associative and R possesses a multiplicative identity 1 which is different from 0.
The distributive law holds: a(b+c)=ab+bc for all a , b , c in G . (The operation · is implied, rather than explicit here.)
A commutative ring is called an integral domain if ab=0 implies a=0 or b=0 (for all a and b in R ).
The proof of Proposition 1.1.1 shows that the multiplicative identity of a commutative ring with identity is unique.
Example 1.1.2: The integers, the rational numbers, and the set of integers modulo n are all commutative rings with identity. The first two are integral domains, but the last is an integral domain if and only if n is a prime. The set of all complex numbers of the form a+bi where a and b are integers and i2=-1 is an integral domain called the ring of Gaussian integers.
Exercise 1.1.3: Working in an integral domain makes it much easier to solve equations. Try to solve (2x-3)(3x-2)=0 for x in the rational numbers; now do the same in the integers modulo 5 and the integers modulo 6.
Proposition 1.1.3: If a is any element of a commutative ring with identity, then a·0=0 .