DOCSLIB.ORG

Notes on Ring Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Notes on Ring Theory Notes on Ring Theory S. F. Ellermeyer Department of Mathematics Kennesaw State University January 15, 2016 Abstract These notes contain an outline of essential definitions and theorems from Ring Theory but only contain a minimal number of examples. Many examples are presented in class and thus it is important to come to class. Also of utmost importance is that the student put alot of effort into doing the assigned homework exercises.The material in these notes is outlined in the order of Chapters 17—33 of Charles Pinter’s “A Book of Abstract Algebra”, which is the textbook we are using in the course. 1Rings A ring is a non—empty set, , along with two operations called addition (usually denoted by the symbol +) and multiplication (usually denoted by the symbol or by no symbol) such that · 1. is an abelian group under addition. (The additive identity element of is usually denoted by 0.) 2. The multiplication operation is associative. 3. Multiplication is distributive over addition, meaning that for all and in we have ( + )= + · · · 1 and ( + ) = + . · · · Letuslookatafewexamplesofrings. Example 1 The trivial ring is the one—element set = 0 with addition operation defined by 0+0 = 0 and multiplication operation de{fi}ned by 0 0=0. · Example 2 The set of integers, , with the standard operations of addition and multiplication is a ring. The set of real numbers, , with the standard operations of addition and multiplication is a ring. The set of rational num- bers, , with the standard operations of addition and multiplication is a ring. The set of complex numbers, , with the standard operations of addition and multiplication is a ring. Example 3 The set 4 = 0 1 2 3 with operations defined by { } + 0 1 2 3 0 1 2 3 · 0 0 1 2 3 0 0 0 0 0 1 1 2 3 0 1 0 1 2 3 2 2 3 0 1 2 0 2 0 2 3 3 0 1 2 3 0 3 2 1 is a ring. (In general, for any given integer 1,theset = 0 1 2 1 with addition and multiplication operations de≥fined “modulo ”isaring.){ − } Example 4 The set, 2 (),ofall2 2 matrices with real entries and operations defined by × + + 1 1 + 2 2 = 1 2 1 2 1 1 2 2 1 + 2 1 + 2 ∙ ¸ ∙ ¸ ∙ ¸ and + + 1 1 2 2 = 1 2 1 2 1 2 1 2 1 1 2 2 12 + 12 12 + 12 ∙ ¸ ∙ ¸ ∙ ¸ is a ring. 2 Example 5 Let () denote the set of all functions from into with operations defined as follows: For and (), + is the function in () defined by ∈ ( + )()= ()+ () for all ∈ and is the function in () defined by · ( )()= () () for all . · ∈ () with these operations is a ring. The additive identity element of a ring, , is usually denoted by the symbol 0 and the additive inverse of any element is usually denoted by .Also,forany and we interpret ∈to mean +( ).The following− Proposition gives some∈ basic algebraic properties− of multiplication− in rings. Proposition 6 If is a ring with additive identity element 0 then 1. 0=0and 0 =0for all . · · ∈ 2. ( )= ( ) and ( ) = ( ) for all and . · − − · − · − · ∈ 3. ( ) ( )= for all and . − · − ∈ Proof. Suppose that is a ring with additive identity element 0. 1. Let .Then ∈ 0= (0 + 0) = 0+ 0. · · · · Since 0+ 0= 0,wecansubtract 0 from both sides of this equation· (which· really· means adding 0 to· both sides) to obtain − · 0+ 0 0= 0 0 · · − · · − · which gives 0+0=0 · which gives 0=0. · The proof that 0 =0is similar. · 3 2. Let and .Then ∈ ( )+ = ( + )= 0=0(by Part 1 which has just been proved). · − · − · Since ( )+ =0and since the additive inverse of is unique (because· − is· a group under addition), then it must be the· case that the additive inverse of is in fact ( ).Inotherwords ( )= ( ). · · − − · · − The proof that ( ) = ( ) is similar. − · − · 3. Let and . Then, by using Part 2 of this Proposition twice, we obtain ∈ ( ) ( )= (( ) )= ( ( )) = . − · − − − · − − · · Note that our definition of the term “ring” above does not require that the multiplication operation of a ring be commutative or that the multipli- cation operation have an identity element in or that (if the multiplication operation does have an identity element) that each element in have a mul- tiplicative inverse. If is a ring for which the multiplication operation is commutative (meaning that = for all and ), then is said to be a commutative ring· .If has· a multiplicative∈ identity element (meaning an element, , such that = = for all ), then is said to be a ring with∈ unity. In any· ring· with unity the multiplica-∈ tive identity must be unique (as is proved in the following proposition). The multiplicative identity element, if it exists, is usually denoted by the symbol 1 andisalsoreferredtoastheunity of . Proposition 7 If is a ring with a multiplicative identity element (a ring with unity), then the multiplicative identity element (the unity) is unique. Proof. Suppose that is a ring with unity and suppose that 1 is a multi- plicative identity element of and that 2 is a multiplicative identity element of .Then 12 = 2 (since 1 is a multiplicative identity) and 12 = 1 (since 2 is a multiplicative identity). 4 We conclude that 1 = 2 and hence that the multiplicative identity is unique. Returningtotheexamplesofringsthatweregivenaboveweseethat The trivial ring is a commutative ring with unity. • , , and are all commutative rings with unity. • 2 () is not a commutative ring but it is a ring with unity. The unity • is 10 = . 01 ∙ ¸ () is a commutative ring with unity. The unity is the function • 1 () defined by 1 ()=1for all . ∈ ∈ If is a ring with unity, then an element is said to be invertible if there exists an element such that =∈ =1.If is an invertible element of a ring with unity,∈,thentheinverseof must be unique (which 1 the reader can prove) and is usually denoted by − . In a non—trivial ring with unity, the additive identity element, 0, is never invertible. The reasoning is as follows: If is a ring with unity and 1=0, then every element must satisfy ∈ = 1= 0=0 · · by Proposition 6. Thus every element of must be equal to 0 and hence is the trivial ring. Stated in contrapositive form this means that if is a non—trivial ring with unity then 1 =0.Nowsupposethat is a non—trivial ring with unity and suppose that 6 0 is invertible. Then there exists such that 0 =1but Proposition 6 then gives 0=1which contradicts what∈ was stated above.· Therefore 0 is not invertible. A commutative ring with unity in which every non—zero element is in- vertible is called a field.Examplesoffields are , , and 5.(More generally, is a field if and only if is a prime number.) , 2 () and () are not fields. Aring,,issaidtohavethecancellation property if for all , and we have ∈ ( = or = )= ( =0or = ) . ⇒ 5 All fields have the cancellation property. also has the cancellation property but 2 () and () do not. Neither does when isnotaprimenumber. An integral domain is a commutative ring with unity which has the cancellation property. Thus all fields are integral domains and is also an integral domain (though it is not a field). If is a ring then an element is called a divisor of zero (or a zero divisor)if =0and there exists∈ such that =0and =0.As 6 ∈ 6 examples note that 2 is a divisor of 0 in the ring 6 because 2 =0and 3 =0 6 6 but 2 3=0(which also means that 3 is a divisor of 0 in 6). In 2 () we can find· many examples of matrices and such that =0and =0 6 6 but =0(the zero matrix). Thus 2 () contains many divisors of zero. In (), the function : defined by ()=2 is a divisor of zero. Why? → Theorem 8 Aring,, has the cancellation property if and only if has no divisors of zero. Proof. Suppose that has the cancellation property. We will use proof by contradiction to show that has no divisors of zero. Let and suppose that is a divisor of 0. Then there exists =0 and there∈ exists with =0such that =0(or =0but we6 will assume without loss∈ of generality6 that =0). Since =0,then = 0 by Proposition 6. The cancellation property then gives us that either =0 or =0. However this is a contradiction to what was deduced above. We conclude that no element of can be a divisor of 0. Next suppose that has not divisors of 0. We will show via a direct proof that must have the cancellation property. Let , and and suppose that = (if we suppose = , the remainder of∈ the proof is handled similarly). Since = ,then +( ()) = 0 and by Proposition 6 we obtain + ( )=0.The distributive− property then gives − ( )=0. − Since has no zero divisors then either =0or =0. That is, either =0or = .Wehavethusshownthat has the cancellation− property. We have definedanintegraldomaintobeacommutativeringwithunity which has the cancellation property. According to the above theorem, it is equivalent to define an integral domain to be a commutative ring with unity which has no divisors of zero. 6.
Load more

Recommended publications
  • Chapter 4 Complex Numbers Course Number
    Chapter 4 Complex Numbers Course Number Section 4.1 Complex Numbers Instructor Objective: In this lesson you learned how to perform operations with Date complex numbers. Important Vocabulary Define each term or concept. Complex numbers The set of numbers obtained by adding real number to real multiples of the imaginary unit i. Complex conjugates A pair of complex numbers of the form a + bi and a – bi. I. The Imaginary Unit i (Page 328) What you should learn How to use the imaginary Mathematicians created an expanded system of numbers using unit i to write complex the imaginary unit i, defined as i = Ö - 1 , because . numbers there is no real number x that can be squared to produce - 1. By definition, i2 = - 1 . For the complex number a + bi, if b = 0, the number a + bi = a is a(n) real number . If b ¹ 0, the number a + bi is a(n) imaginary number .If a = 0, the number a + bi = bi is a(n) pure imaginary number . The set of complex numbers consists of the set of real numbers and the set of imaginary numbers . Two complex numbers a + bi and c + di, written in standard form, are equal to each other if . and only if a = c and b = d. II. Operations with Complex Numbers (Pages 329-330) What you should learn How to add, subtract, and To add two complex numbers, . add the real parts and the multiply complex imaginary parts of the numbers separately. numbers Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company.
    [Show full text]
  • Comultiplication on Monoids
    Internat. J. Math. & Math. Sci. 803 VOL. 20 NO. 4 (1997) 803-812 COMULTIPLICATION ON MONOIDS MARTIN ARKOWITZ Mathematics Department Dartmouth College Hanover, NH 03755 E-mail address: [email protected] MAURIClO GUTIERREZ Mathematics Department Tufts University Medford, MA 02155 E-mail address: [email protected] (Received April 14, 1997) ABSTRACT. A comultiplication on a monoid S is a homomorphism rn: S S S (the free product of S with itself) whose composition with each projection is the identity homomorphism. We investigate how the existence of a comultiplication on S restricts the structure of S. We show that a monoid which satisfies the inverse property and has a comultiplication is cancellative and equidivisible. Our main result is that a monoid S which satisfies the inverse property admits a comultiplication if and only if S is the free product of a free monoid and a free group. We call these monoids semi-free and we study different comultiplications on them. KEY WORDS AND PHRASES: Monoid, free product, comultiplication, inverse property. cancellative monoid, equidivisibility, semi-free monoid. 1991 AMS SUBJECT CLASSIFICATION CODES: Primary: 20M05, 20M10 Secondary: 20M99, 20N99. 1. INTRODUCTION Comultiplications can be defined for objects in any category with coproducts and zero mor- phisms. Given such an object X, a comultiplication is a morphism m X X II X (the coproduct of X with itself), such that the composition of rn with either projection X t X X is the identity morphism. The general theory has been surveyed in [1], and in [3] it is specialized to algebraic systems. In [2], [5] and [6], comultiplications on groups have been studied.
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set)
    LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most often name sets using capital letters, like A, B, X, Y , etc., while the elements of a set will usually be given lower-case letters, like x, y, z, v, etc. Two sets X and Y are called equal if X and Y consist of exactly the same elements. In this case we write X = Y . Example 1 (Examples of Sets). (1) Let X be the collection of all integers greater than or equal to 5 and strictly less than 10. Then X is a set, and we may write: X = f5; 6; 7; 8; 9g The above notation is an example of a set being described explicitly, i.e. just by listing out all of its elements. The set brackets {· · ·} indicate that we are talking about a set and not a number, sequence, or other mathematical object. (2) Let E be the set of all even natural numbers. We may write: E = f0; 2; 4; 6; 8; :::g This is an example of an explicity described set with infinitely many elements. The ellipsis (:::) in the above notation is used somewhat informally, but in this case its meaning, that we should \continue counting forever," is clear from the context. (3) Let Y be the collection of all real numbers greater than or equal to 5 and strictly less than 10. Recalling notation from previous math courses, we may write: Y = [5; 10) This is an example of using interval notation to describe a set.
    [Show full text]
  • Reachability Problems in Quaternion Matrix and Rotation Semigroups
    Reachability Problems in Quaternion Matrix and Rotation Semigroups Paul Bell , Igor Potapov ∗ Department of Computer Science, University of Liverpool, Ashton Building, Ashton St, Liverpool L69 3BX, U.K. Abstract We examine computational problems on quaternion matrix and rotation semigroups. It is shown that in the ultimate case of quaternion matrices, in which multiplication is still associative, most of the decision problems for matrix semigroups are un- decidable in dimension two. The geometric interpretation of matrix problems over quaternions is presented in terms of rotation problems for the 2 and 3-sphere. In particular, we show that the reachability of the rotation problem is undecidable on the 3-sphere and other rotation problems can be formulated as matrix problems over complex and hypercomplex numbers. Key words: Quaternions, Matrix semigroups, Rotation semigroups, Membership problem, Undecidability, Post’s correspondence problem 1 Introduction Quaternions have long been used in many fields including computer graphics, robotics, global navigation and quantum physics as a useful mathematical tool for formulating the composition of arbitrary spatial rotations and establishing the correctness of algorithms founded upon such compositions. Many natural questions about quaternions are quite difficult and correspond to fundamental theoretical problems in mathematics, physics and computational theory. Unit quaternions actually form a double cover of the rotation group ∗ Corresponding author. Email addresses: [email protected] (Paul Bell), [email protected] (Igor Potapov). Preprint submitted to Elsevier 4 July 2008 SO3, meaning each element of SO3 corresponds to two unit quaternions. This makes them expedient for studying rotation and angular momentum and they are particularly useful in quantum mechanics.
    [Show full text]
  • Comultiplication on Monoids
    Internat. J. Math. & Math. Sci. 803 VOL. 20 NO. 4 (1997) 803-812 COMULTIPLICATION ON MONOIDS MARTIN ARKOWITZ Mathematics Department Dartmouth College Hanover, NH 03755 E-mail address: [email protected] MAURIClO GUTIERREZ Mathematics Department Tufts University Medford, MA 02155 E-mail address: [email protected] (Received April 14, 1997) ABSTRACT. A comultiplication on a monoid S is a homomorphism rn: S S S (the free product of S with itself) whose composition with each projection is the identity homomorphism. We investigate how the existence of a comultiplication on S restricts the structure of S. We show that a monoid which satisfies the inverse property and has a comultiplication is cancellative and equidivisible. Our main result is that a monoid S which satisfies the inverse property admits a comultiplication if and only if S is the free product of a free monoid and a free group. We call these monoids semi-free and we study different comultiplications on them. KEY WORDS AND PHRASES: Monoid, free product, comultiplication, inverse property. cancellative monoid, equidivisibility, semi-free monoid. 1991 AMS SUBJECT CLASSIFICATION CODES: Primary: 20M05, 20M10 Secondary: 20M99, 20N99. 1. INTRODUCTION Comultiplications can be defined for objects in any category with coproducts and zero mor- phisms. Given such an object X, a comultiplication is a morphism m X X II X (the coproduct of X with itself), such that the composition of rn with either projection X t X X is the identity morphism. The general theory has been surveyed in [1], and in [3] it is specialized to algebraic systems. In [2], [5] and [6], comultiplications on groups have been studied.
    [Show full text]
  • Algebraic Systems, Spring 2014, January, 2014 Edition Gabriel Kerr
    Algebraic Systems, Spring 2014, January, 2014 Edition Gabriel Kerr Contents Chapter 0. Peano Axioms for Natural Numbers - An Introduction to Proofs 5 0.1. Sets and Logic 5 Exercises 7 0.2. Peano Axioms 8 Exercises 15 0.3. Relations 15 Exercises 19 Chapter 1. Basic Arithmetic 21 1.1. The set Z 21 Exercises 23 1.2. The ring Z 23 Exercises 27 1.3. Factoring integers, part I 27 Exercises 30 1.4. Factoring integers, part II 30 Exercises 32 1.5. Modular Arithmetic 32 Exercises 34 1.6. The fields Q and R 34 Exercises 39 1.7. The field C 39 Exercises 40 3 CHAPTER 0 Peano Axioms for Natural Numbers - An Introduction to Proofs We begin this course with the construction of the natural number system. The Peano Axioms form the heart of much of mathematics and lay the foundation for algebra and analysis. Every attempt will be made to stay away from unnecessary abstraction. This should be a guiding principle when working out the exercises as well! 0.1. Sets and Logic This section should serve as a very short introduction to 20th century mathe- matics. That is, it is an introduction to proofs involving sets. Set theory is in fact a subject within itself that was initiated and studied by many in the early 20th century. This was in response to several paradoxes that had come up. For us, a set is a collection of things, called elements. For example, Atoms could be the set of all atoms in the universe while Adams could be the set of all people named Adam in your family.
    [Show full text]
  • Vector Spaces
    Chapter 1 Vector Spaces Linear algebra is the study of linear maps on finite-dimensional vec- tor spaces. Eventually we will learn what all these terms mean. In this chapter we will define vector spaces and discuss their elementary prop- erties. In some areas of mathematics, including linear algebra, better the- orems and more insight emerge if complex numbers are investigated along with real numbers. Thus we begin by introducing the complex numbers and their basic✽ properties. 1 2 Chapter 1. Vector Spaces Complex Numbers You should already be familiar with the basic properties of the set R of real numbers. Complex numbers were invented so that we can take square roots of negative numbers. The key idea is to assume we have i − i The symbol was√ first a square root of 1, denoted , and manipulate it using the usual rules used to denote −1 by of arithmetic. Formally, a complex number is an ordered pair (a, b), the Swiss where a, b ∈ R, but we will write this as a + bi. The set of all complex mathematician numbers is denoted by C: Leonhard Euler in 1777. C ={a + bi : a, b ∈ R}. If a ∈ R, we identify a + 0i with the real number a. Thus we can think of R as a subset of C. Addition and multiplication on C are defined by (a + bi) + (c + di) = (a + c) + (b + d)i, (a + bi)(c + di) = (ac − bd) + (ad + bc)i; here a, b, c, d ∈ R. Using multiplication as defined above, you should verify that i2 =−1.
    [Show full text]
  • List of Mathematical Symbols by Subject from Wikipedia, the Free Encyclopedia
    List of mathematical symbols by subject From Wikipedia, the free encyclopedia This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology. The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles. Contents 1 Guide 2 Set theory 2.1 Definition symbols 2.2 Set construction 2.3 Set operations 2.4 Set relations 2.5 Number sets 2.6 Cardinality 3 Arithmetic 3.1 Arithmetic operators 3.2 Equality signs 3.3 Comparison 3.4 Divisibility 3.5 Intervals 3.6 Elementary functions 3.7 Complex numbers 3.8 Mathematical constants 4 Calculus 4.1 Sequences and series 4.2 Functions 4.3 Limits 4.4 Asymptotic behaviour 4.5 Differential calculus 4.6 Integral calculus 4.7 Vector calculus 4.8 Topology 4.9 Functional analysis 5 Linear algebra and geometry 5.1 Elementary geometry 5.2 Vectors and matrices 5.3 Vector calculus 5.4 Matrix calculus 5.5 Vector spaces 6 Algebra 6.1 Relations 6.2 Group theory 6.3 Field theory 6.4 Ring theory 7 Combinatorics 8 Stochastics 8.1 Probability theory 8.2 Statistics 9 Logic 9.1 Operators 9.2 Quantifiers 9.3 Deduction symbols 10 See also 11 References 12 External links Guide The following information is provided for each mathematical symbol: Symbol: The symbol as it is represented by LaTeX.
    [Show full text]
  • 24 Rings: Definition and Basic Results
    Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 24 Rings: Definition and Basic Results In this section, we introduce another type of algebraic structure, called ring. A group is an algebraic structure that requires one binary operation. A ring is an algebraic structure that requires two binary operations that satisfy some conditions listed in the following definition. Definition 24.1 A ring is a nonempty set R with two binary operations (usually written as ad- dition and multiplication) such that for all a; b; c 2 R; (1) R is closed under addition: a+b 2 R: (2) Addition is associative: (a+b)+c = a + (b + c). (3) Addition is commutative: a + b = b + a: (4) R contains an additive identity element, called zero and usually denoted by 0 or 0R: a + 0 = 0 + a = a: (5) Every element of R has an additive inverse: a + (¡a) = (¡a) + a = 0: (6) R is closed under multiplication: ab 2 R: (7) Multiplication is associative: (ab)c = a(bc): (8) Multiplication distributes over addition: a(b + c) = ab + ac and (a + b)c = ac + bc: If ab = ba for all a; b 2 R then we call R a commutative ring. In other words, a ring is a commutative group with the operation + and an additional operation, multiplication, which is associative and is distributive with respect to +. Remark 24.1 Note that we don’t require a ring to be commutative with respect to multiplica- tion, or to have multiplicative identity, or to have multiplicative inverses. A ring may have these properties, but is not required to.
    [Show full text]
  • Vector Spaces
    Vector Spaces Math 240 Definition Properties Set notation Subspaces Vector Spaces Math 240 | Calculus III Summer 2013, Session II Wednesday, July 17, 2013 Vector Spaces Agenda Math 240 Definition Properties Set notation Subspaces 1. Definition 2. Properties of vector spaces 3. Set notation 4. Subspaces Vector Spaces Motivation Math 240 Definition Properties Set notation Subspaces We know a lot about Euclidean space. There is a larger class of n objects that behave like vectors in R . What do all of these objects have in common? Vector addition a way of combining two vectors, u and v, into the single vector u + v Scalar multiplication a way of combining a scalar, k, with a vector, v, to end up with the vector kv A vector space is any set of objects with a notion of addition n and scalar multiplication that behave like vectors in R . Vector Spaces Examples of vector spaces Math 240 Definition Real vector spaces Properties n Set notation I R (the archetype of a vector space) Subspaces I R | the set of real numbers I Mm×n(R) | the set of all m × n matrices with real entries for fixed m and n. If m = n, just write Mn(R). I Pn | the set of polynomials with real coefficients of degree at most n I P | the set of all polynomials with real coefficients k I C (I) | the set of all real-valued functions on the interval I having k continuous derivatives Complex vector spaces n I C, C I Mm×n(C) Vector Spaces Definition Math 240 Definition Properties Definition Set notation A vector space consists of a set of scalars, a nonempty set, V , Subspaces whose elements are called vectors, and the operations of vector addition and scalar multiplication satisfying 1.
    [Show full text]
  • Introduction to Groups, Rings and Fields
    Introduction to Groups, Rings and Fields HT and TT 2011 H. A. Priestley 0. Familiar algebraic systems: review and a look ahead. GRF is an ALGEBRA course, and specifically a course about algebraic structures. This introduc- tory section revisits ideas met in the early part of Analysis I and in Linear Algebra I, to set the scene and provide motivation. 0.1 Familiar number systems Consider the traditional number systems N = 0, 1, 2,... the natural numbers { } Z = m n m, n N the integers { − | ∈ } Q = m/n m, n Z, n = 0 the rational numbers { | ∈ } R the real numbers C the complex numbers for which we have N Z Q R C. ⊂ ⊂ ⊂ ⊂ These come equipped with the familiar arithmetic operations of sum and product. The real numbers: Analysis I built on the real numbers. Right at the start of that course you were given a set of assumptions about R, falling under three headings: (1) Algebraic properties (laws of arithmetic), (2) order properties, (3) Completeness Axiom; summarised as saying the real numbers form a complete ordered field. (1) The algebraic properties of R You were told in Analysis I: Addition: for each pair of real numbers a and b there exists a unique real number a + b such that + is a commutative and associative operation; • there exists in R a zero, 0, for addition: a +0=0+ a = a for all a R; • ∈ for each a R there exists an additive inverse a R such that a+( a)=( a)+a = 0. • ∈ − ∈ − − Multiplication: for each pair of real numbers a and b there exists a unique real number a b such that · is a commutative and associative operation; • · there exists in R an identity, 1, for multiplication: a 1 = 1 a = a for all a R; • · · ∈ for each a R with a = 0 there exists an additive inverse a−1 R such that a a−1 = • a−1 a = 1.∈ ∈ · · Addition and multiplication together: forall a,b,c R, we have the distributive law a (b + c)= a b + a c.
    [Show full text]