This document provides an abstract for a course on Abstract Algebra taught by Gemma P. Salasalan. It includes the following key points:
- The course will cover topics like sets, arithmetic in the integers, congruence and congruence classes, and relations and operations. It will use the textbooks "Algebra" by Hungerford and "Fundamental Abstract Algebra" by Malik.
- Definitions and examples are provided for fundamental concepts in abstract algebra like sets, the division algorithm, divisibility, greatest common divisors, and congruence modulo n.
- Properties of binary relations like reflexivity, symmetry, and transitivity are defined.
Mathematicians over the last two centuries have been used to the idea of considering a collection of objects/numbers as a single entity. These entities are what are typically called sets. The technique of using the concept of a set to answer questions is hardly new. It has been in use since ancient times. However, the rigorous treatment of sets happened only in the 19-th century due to the German mathematician Georg Cantor. He was solely responsible in ensuring that sets had a home in mathematics. Cantor developed the concept of the set during his study of the trigonometric series, which is now known as the limit point or the derived set operator. He developed two types of transfinite numbers, namely, transfinite ordinals and transfinite cardinals. His new and path-breaking ideas were not well received by his contemporaries. Further, from his definition of a set, a number of contradictions and paradoxes arose. One of the most famous paradoxes is the Russellβs Paradox, due to Bertrand Russell in 1918. This paradox amongst others, opened the stage for the development of axiomatic set theory. The interested reader may refer to Katz [
Sets & Set Operation
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 11, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Mathematicians over the last two centuries have been used to the idea of considering a collection of objects/numbers as a single entity. These entities are what are typically called sets. The technique of using the concept of a set to answer questions is hardly new. It has been in use since ancient times. However, the rigorous treatment of sets happened only in the 19-th century due to the German mathematician Georg Cantor. He was solely responsible in ensuring that sets had a home in mathematics. Cantor developed the concept of the set during his study of the trigonometric series, which is now known as the limit point or the derived set operator. He developed two types of transfinite numbers, namely, transfinite ordinals and transfinite cardinals. His new and path-breaking ideas were not well received by his contemporaries. Further, from his definition of a set, a number of contradictions and paradoxes arose. One of the most famous paradoxes is the Russellβs Paradox, due to Bertrand Russell in 1918. This paradox amongst others, opened the stage for the development of axiomatic set theory. The interested reader may refer to Katz [
Sets & Set Operation
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 11, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
These slides are a summary of the Well-Ordering Principle.
Video explains these slides is available in this link
https://youtu.be/EkleZiBtYyk
Reference books for these slides are
A Transition to Advanced Mathematics 8th Edition,
by Douglas Smith, Maurice Eggen, Richard St. Andre. ISBN-13: 978-1285463261, published by Cengage Learning (August 6, 2014).
https://www.cengagebrain.co.uk/shop/isbn/9781285463261
and
Discrete Mathematics with Applications, 3nd Edition, (1995)
By Susanna S. Epp, ISBN13: 9780534359454,
published by Thomson-Brooks/Cole Publishing Company.
Solution Manual for First Course in Abstract Algebra A, 8th Edition by John B...ssifa0344
Β
Solution Manual for First Course in Abstract Algebra A, 8th Edition by John B. Fraleigh, Verified Chapters 1 - 56, Complete Newest Version.pdf
Solution Manual for First Course in Abstract Algebra A, 8th Edition by John B. Fraleigh, Verified Chapters 1 - 56, Complete Newest Version.pdf
These slides are a summary of the Well-Ordering Principle.
Video explains these slides is available in this link
https://youtu.be/EkleZiBtYyk
Reference books for these slides are
A Transition to Advanced Mathematics 8th Edition,
by Douglas Smith, Maurice Eggen, Richard St. Andre. ISBN-13: 978-1285463261, published by Cengage Learning (August 6, 2014).
https://www.cengagebrain.co.uk/shop/isbn/9781285463261
and
Discrete Mathematics with Applications, 3nd Edition, (1995)
By Susanna S. Epp, ISBN13: 9780534359454,
published by Thomson-Brooks/Cole Publishing Company.
Solution Manual for First Course in Abstract Algebra A, 8th Edition by John B...ssifa0344
Β
Solution Manual for First Course in Abstract Algebra A, 8th Edition by John B. Fraleigh, Verified Chapters 1 - 56, Complete Newest Version.pdf
Solution Manual for First Course in Abstract Algebra A, 8th Edition by John B. Fraleigh, Verified Chapters 1 - 56, Complete Newest Version.pdf
How to Split Bills in the Odoo 17 POS ModuleCeline George
Β
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Model Attribute Check Company Auto PropertyCeline George
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In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Palestine last event orientationfvgnh .pptxRaedMohamed3
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An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
How to Make a Field invisible in Odoo 17Celine George
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It is possible to hide or invisible some fields in odoo. Commonly using βinvisibleβ attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Operation βBlue Starβ is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
1. Abstract Algebra
Abstract Algebra
Gemma P. Salasalan,Ph.D
Institute of Arts and Sciences
Davao del Sur State College
Matti, Digos City
January 2022
Gemma P. Salasalan,Ph.D Abstract Algebra
2. Abstract Algebra
Textbook: Algebra by Hungerford
References:
1. Fundamental Abstract Algebra by Malik
2. Abstract Algebra by Fraleigh
Gemma P. Salasalan,Ph.D Abstract Algebra
4. Abstract Algebra
Sets
Definition 1.1.
A set is a collection of objects. A set S with only a finite number of
elements is called a finite set; otherwise S is called an infinite set. We
let |S| denote the number of elements of S. We denote a finite set by a
listing of its elements within braces {}.
Given a set S, we use the notation x β S and x < S to mean x is a member
{element} of S and x is not a member {element} of S, respectively.
Definition 1.2.
A set A is said to be a subset of a set S if every element of A is an
element of S. In this case, we write A β S and say that A is contained in
S. If A β S, but A , S, then we write A β S and say that A is a proper
subset of S.
Gemma P. Salasalan,Ph.D Abstract Algebra
6. Abstract Algebra
Sets
Definition 1.4.
The null set or empty set is the set with no elements. We usually
denote the empty set by β . For any set A, we have β β A.
Given a set S, the notation
A = {x β S|P(x)}
Definition 1.5.
Definition 1.6.
Gemma P. Salasalan,Ph.D Abstract Algebra
12. Abstract Algebra
Sets
Example 1.12.
Let (x, y), (z, w) β A Γ B. Show that (x, y) = (z, w) if and only if x = z
and y = w.
Gemma P. Salasalan,Ph.D Abstract Algebra
13. Abstract Algebra
Sets
Definition 1.13.
For a subset A of a set S, let Aβ²
denote the subset SA. Aβ²
is called the
complement of A in S.
Gemma P. Salasalan,Ph.D Abstract Algebra
14. Abstract Algebra
Arithmetic in Z
Definition 2.1 (Well-ordering Principle).
Every nonempty subset of the set of nonnegative integers contains a
smallest element.
Example 2.2.
1. The set of Natural Numbers N is a well-ordered set.
2. The open interval (0, 2) is a non-empty subset of R but it has no
smallest element.
Consider the following grade-school division problem:
Gemma P. Salasalan,Ph.D Abstract Algebra
15. Abstract Algebra
Arithmetic in Z
Denote aβ dividend, bβ divisor, qβ quotient and rβ remainder
Theorem 2.3 (The Division Algorithm).
Let a, b be integers with b > 0. Then there exist unique integers q and r
such that
a = bq + r and 0 β€ r < b.
Example 2.4.
Gemma P. Salasalan,Ph.D Abstract Algebra
16. Abstract Algebra
Arithmetic in Z
Proof :
Suppose a, b β Z and b > 0. Consider
S = {a β bq|q β Z and a β d β₯ 0}.
We need to show that:
1. S , β .
2. By WOP, we can find a least element r β r < b.
2. Uniqueness, that is, r and q are unique.
1. Consider the following cases:
Case 1. a β₯ 0.
We take q = 0, then a β bq = a β b(0) = a . This implies that a β S.
Case 2. a < 0 .
We take q = a, then a β b(a) = a(1 β b). Note that a < 0 , b > 0 (from
assumption), and a(1 β b) β₯ 0 . This implies that a β b(a) β S.
Thus, S , β . By WOP, S has a least element r = a β bq for some integer
q. Hence, a = bq + r and r β₯ 0.
Gemma P. Salasalan,Ph.D Abstract Algebra
17. Abstract Algebra
Arithmetic in Z
Proof : cont...
2. Suppose r β₯ b. Then r = b + rβ , where 0 β€ rβ²
< r. So,
a = bq + r = bq + b + rβ²
= (q + 1)b + rβ²
, so that rβ²
= a β (q + 1)b is an element of S smaller than r. This
contradicts the fact that r is the least element of S. Thus, r < d.
3. Suppose that there are integers q1 and r1 such that a = bq1 + r1 and
0 β€ r1 < b. We need to show that, q1 = q and r1 = r.
Since a = bq + r and a = bq1 + r1, we have
bq + r = bq1 + r1
so,
b(q β q1) = r1 β r (1)
Note that,
0 β€ r < b (2)
0 β€ r1 < b (3)
Gemma P. Salasalan,Ph.D Abstract Algebra
18. Abstract Algebra
Arithmetic in Z
Multiplying β1 to inequality (2), we obtain
βb β€ βr < 0 (4)
0 β€ r1 < b (2)
Adding these two inequalities
βb < r1 β r β€ b
βb < b(q β q1) < b from Equation (1)
β1 < q β q1 < 1 Divide each term by b
But q β q1 is an integer (because q and q1 are integers) and the only
integer strictly between -1 and 1 is 0. Therefore q β q1 = 0 and q = q1.
Substituting q β q1 = 0 in Equation (1),
b(q β q1) = r1 β r (1)
b(0) = r1 β r
0 = r1 β r
r1 = r.
Thus the quotient and remainder are unique.
Gemma P. Salasalan,Ph.D Abstract Algebra
19. Abstract Algebra
Arithmetic in Z
Definition 2.5 (Divisibility).
Let a and b be integers with b , 0. We say that b divides a (or that b is a
divisor of a, or that b is a factor of a) if a = bc for some integer c. In
symbols,βb divides aβ is written b|a and βb does not divide aβ is written
b β€ a.
Example 2.6.
3|24 because 24 = 3 β’ 8, but 3 β€ 17. Negative divisors are allowed: β6|54
because 54 = (β6)(β9), but β6 β€ β13.
Gemma P. Salasalan,Ph.D Abstract Algebra
20. Abstract Algebra
Arithmetic in Z
Remark 2.7.
1. If b divides a, then a = bc for some c. Hence βa = b(βc), so that
b|(βa). An analogous argument shows that every divisor of βa is also
a divisor of a. Therefore, a and βa have the same divisors.
2. Suppose a , 0 and b|a. Then a = bc, so that |a| = |b||c|.
Consequently, 0 β€ |b| β€ |a|. This last inequality is equivalent to
β|a| β€ |b| β€ |a| . Therefore
i. every divisor of the nonzero integer a is less than or equal to |a|;
ii. nonzero integer has only finitely many divisors.
Gemma P. Salasalan,Ph.D Abstract Algebra
21. Abstract Algebra
Arithmetic in Z
Definition 2.8.
Let a and b be integers, not both 0. The greatest common divisor
(gcd) of a and b is the largest integer d that divides both a and b. In other
words, d is the gcd of a and b provided that
1. d|a and d|b;
2. if c|a and c|b, then c β€ d.
The greatest common divisor of a and b is usually denoted GCD(a, b).
Definition 2.9.
If p is an integer greater than 1, then p is a prime number if the only
divisors of p are 1 and p. A positive integer greater than 1 that is not a
prime number is called composite.
Example 2.10.
The greatest common divisor of 12 and 30 is 6, that is, GCD(12, 30) = 6.
The only common divisors of 10 and 21 are 1 and β1. Hence
GCD(10, 21) = 1. Two integers whose greatest common divisor is 1, such
as 10 and 21, are said to be relatively prime.
Gemma P. Salasalan,Ph.D Abstract Algebra
22. Abstract Algebra
Arithmetic in Z
Example 2.11.
Here are some examples to illustrate the definitions above.
1. GCD(45, 60) = 15, since 45 = 15 β’ 3 and 60 = 15 β’ 4 and 15 is the
largest number that divides both 45 and 60.
2. 45 and 60 are not relatively prime.
3. 45 and 16 are relatively prime since GCD(45, 16) = 1.
Gemma P. Salasalan,Ph.D Abstract Algebra
23. Abstract Algebra
Arithmetic in Z
Theorem 2.12.
Let a and b be integers, not both 0, and let d be their greatest common
divisor. Then there exist (not necessarily unique) integers u and v such
that d = au + bv.
Proof :
Let S be the set of all linear combinations of a and b, that is
S = {am + bn|m, n β Z}.
Step 1. Find the smallest positive element of S.
Note that a2
+ b2
= aa + bb is in S and a2
+ b2
β₯ 0. Since a and b are not
both 0, a2
+ b2
must be positive. By WOP, S contains positive integer
and hence, contains a least element. Let t be the smallest positive integer
in S. Hence, we can write t = au + bv for some integers u and v.
Gemma P. Salasalan,Ph.D Abstract Algebra
24. Abstract Algebra
Arithmetic in Z
Proof : cont... Step 2. Prove that t is the gcd of a and b, that is, t = d
We must prove that t satisfies the two conditions in the definition of the
gcd:
1. t|a and t|b;
2. if c|a and c|b, then c β€ t.
Proof of (1): By the Division Algorithm, there are integers q and r such
that
a = tq + r , with 0 β€ r t.
Consequently,
r = a β tq, (1)
r = a β (au + bv)q = a β aqu β bvq, (2)
r = a(1 β qu) + b(βvq) (3)
Thus r is a linear combination of a and b, and hence r β S.
Gemma P. Salasalan,Ph.D Abstract Algebra
25. Abstract Algebra
Arithmetic in Z
Since r t (the smallest positive element of S), we know that r is not
positive. Since r β₯ 0, the only possibility is that r = 0. Therefore,
a = tq + r = tq + 0 = tq so that t|a.
Similarly, t|b. Hence, t is a common divisor of a and b.
Proof of (2): Let c be any other common divisor of a and b, so that c|a
and c|b. Then a = ck and b = cs for some integers k and s. Consequently,
t = au + bv = (ck)u + (cs)v (4)
= c(ku + sv). (5)
This implies that c|t. Hence, c β€ |t| by Remark 2.7 . But t is positive, so
|t| = t. Thus, c β€ t. This implies that t is the greatest common divisor d.
Gemma P. Salasalan,Ph.D Abstract Algebra
26. Abstract Algebra
Arithmetic in Z
Corollary 2.13.
Let a and b be integers, not both 0, and let d be a positive integer. Then d
is the greatest common divisor of a and b if and only if d satisfies these
conditions:
i. d|a and d|b;
i. if c|a and c|b, then c|d.
Gemma P. Salasalan,Ph.D Abstract Algebra
27. Abstract Algebra
Arithmetic in Z
Theorem 2.14.
If a|bc and (a, b) = 1, then a|c.
Proof :
Since (a, b) = 1, Theorem 2.12 shows that au + bv = 1 for some integers u
and v. Multiplying this equation by c shows that acu + bcv = c. But a|bc,
so that bc = ar for some r. Therefore,
c = acu + bcv = acu + (ar)v = a(cu + ro).
This shows that a|c.
Gemma P. Salasalan,Ph.D Abstract Algebra
28. Abstract Algebra
Arithmetic in Z
Congruence and Congruence Classes
Congruence and Congruence
Classes
Gemma P. Salasalan,Ph.D Abstract Algebra
29. Abstract Algebra
Arithmetic in Z
Congruence and Congruence Classes
Definition 2.15.
Let a,b, n be integers with n 0. Then a is congruent to b modulo n
[written βa = b(modn)β], provided that n divides a β b.
Example 2.16.
1. 17 = 5(mod6) because 6 divides 17 β 5 = 12.
2. 4 = 25(mod7) because 7 divides 4 β 25 = β21.
Gemma P. Salasalan,Ph.D Abstract Algebra
30. Abstract Algebra
Arithmetic in Z
Relation and Operations
Relation and Operations
Gemma P. Salasalan,Ph.D Abstract Algebra
31. Abstract Algebra
Arithmetic in Z
Relation and Operations
Definition 2.17.
A relation R on a set S (more precisely, a binary relation on S, since it
will be a relation between pairs of elements of S) is a subset of S x S.
Example 2.18.
Let S = {2, 3, 5, 6} and let R mean βdivides.β Since
2R2, 2R6, 3R3, 3R6, 5R5, 6R6, we have
R = {(2, 2), (6, 2), (3, 3), (6, 3), (5, 5), (6, 6)}.
PROPERTIES OF BINARY RELATIONS
Let R be a relation on a set S and a, b, c β S. Then
1. R is called reflexive if aRa for every a β S.
2. R is called symmetric if whenever aRb then bRa.
2. R is called transitive if whenever aRb and bRc then aRc.
Gemma P. Salasalan,Ph.D Abstract Algebra
32. Abstract Algebra
Arithmetic in Z
Relation and Operations
Example 2.19.
1. Let R be the set of real numbers and R mean βis less than or equal
to.β Thus, any number is less than or equal to itself so R is reflexive.
2. Let R be the set of real numbers and R mean βis less than or equal
to.β Now 3 is less than or equal to 5 but 5 is not less than or equal to
3. Hence R is not symmetric.
β’ Let R be the set of real numbers and R mean βis less than or equal
to.β If x β€ y and y β€ z, then x β€ z. Hence, R is transitive.
Definition 2.20.
A relation R on a set S is called an equivalence relation on S when R is
reflexive, symmetric and transitive.
Gemma P. Salasalan,Ph.D Abstract Algebra
34. Abstract Algebra
Arithmetic in Z
Relation and Operations
Definition 2.21 (Equivalence set or Equivalence class).
Let S be a set and R be an equivalence relation on S. If a β S, the
elements y β S satisfying yRa constitute a subset, [a], of S, called an
equivalence set or equivalence class.
Thus, formally,
[a] = {y : y β S, yRa}
Gemma P. Salasalan,Ph.D Abstract Algebra
36. Abstract Algebra
Arithmetic in Z
Relation and Operations
Definition 2.23 (Partition).
A set {A, B, C, ...} of non empty subsets of a set S will be called a
partition of S provided
1. A βͺ B βͺ Cβͺ, ... = S
2. The intersection of every pair of distinct subsets is the empty set.
Theorem 2.24.
Gemma P. Salasalan,Ph.D Abstract Algebra
37. Abstract Algebra
Arithmetic in Z
Relation and Operations
Example 2.25.
Let a relation βΌ be defined on the set R of real numbers by xRy if and
only if |x| = |y|. Show that βΌ is an equivalence relation.
Solution:
1. Let a β R. Then |a| = |a|. So a βΌ a and R is reflexive.
2. Let a, b β R and suppose that a βΌ b. Then |a| = |b|. So |b| = |a|.
Thus, b βΌ a and R is symmetric.
3. Let a, b, c β R and suppose that a βΌ b and b βΌ c. Then |a| = |b| and
|b| = |c|. So |a| = |c|. Thus, a βΌ c and R is transitive.
Therefore, R is an equivalence relation on R.
Gemma P. Salasalan,Ph.D Abstract Algebra
38. Abstract Algebra
Arithmetic in Z
Relation and Operations
Example 2.26.
Consider the set S = R where x βΌ y if and only if x2
= y2
. Prove that βΌ
is an equivalence relation on S.
Proof :
Let x, y, z β S.
i. Since x2
= x2
β x β S. So, x βΌ x and βΌ is reflexive.
ii. Suppose x βΌ y. Then x2
= y2
.
Note that y2
= x2
. So, y βΌ x and βΌ is symmetric.
iii. Suppose x βΌ y and y βΌ z. Then x2
= y2
and y2
= z2
.
x2
= y2
β = z2
Thus, x βΌ z and βΌ is an equivalence relation on S.
Gemma P. Salasalan,Ph.D Abstract Algebra
40. Abstract Algebra
Lesson 2
Operations
Definition 4.1.
A binary operation β * β on a non-empty set S is a mapping which
associates with each ordered pair (a, b) of elements of S a uniquely defined
element a β b of S.
A binary operation on a set S is a mapping of S Γ S into S.
β : S Γ S β S such that a β b β S, β a, b β S
Example 4.2.
Let S = 0, 1, 2, 3, 4. Neither addition or multiplication are binary operation
on set S. Since 2 + 3 = 5 S and 2 Β· 3 = 6 S.
Gemma P. Salasalan,Ph.D Abstract Algebra
41. Abstract Algebra
Lesson 2
Operations
Properties of Binary Operation
i. Closure
A binary operation β on a non-empty set S has closure property, if
a β S, b β S β a β b β S.
ii. Associative
The associative property of binary operations holds if, for a
non-empty set S, we can write (a β b) β c = a β (b β c), where a, b, c β S.
iii. Commutative
A binary operation * on a non-empty set S is commutative, if
a β b = b β a, for all (a, b) β S.
iv. Identity
A set S is said to have an identity element with respect to a binary
operation * on S if there exists an element e β S with the property
e β x = x β e = x for every x β S:
Gemma P. Salasalan,Ph.D Abstract Algebra
42. Abstract Algebra
Lesson 2
Operations
iv. Inverse
Consider a set S having the identity element e with respect to a
binary operation *. An element y β S is called an inverse of x β S
provided that, x β y = y β x = e. We can write, y = aβ1
.
Definition 4.3.
Gemma P. Salasalan,Ph.D Abstract Algebra