DOCSLIB.ORG

Systems of Linear Equations Math 130 Linear Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Systems of Linear Equations Math 130 Linear Algebra A general m × n system looks like this: a11x1 + a12x2 + ··· + a1nxn = b1 a21x1 + a22x2 + ··· + a2nxn = b2 . am1x1 + am2x2 + ··· + amnxn = bm Systems of Linear Equations where the n variables are x1; x2; : : : ; xn. The m Math 130 Linear Algebra constants that appear on the right side of the m D Joyce, Fall 2015 equations are denoted b1; b2; : : : ; bm. Since each of the m equations has n coefficients for the n vari- ables, there are mn coefficients in all, and these are You already know how to solve systems of linear denoted with doubly subscripts, so, for instance a equations, and we've looked at the ancient Chinese 35 is the coefficient of the variable x in the third equa- method of elimination to solve them. We'll analyze 5 tion. that algorithm and introduce some terminology as- A solution to the system is an n-tuple of values sociated with it. for the n variables that make all the equations si- Although we'll have a lot more use for matrices multaneously true. For example, in the 3×2 system later, for the time being we'll use them as a short- above, a solution is (x; y) = (2; 1). In fact, it's the hand notation when describing systems of linear only solution for that system. equations. Matrices. There are two different matrices we'll Standard notation and terminology. When use to analyze systems of linear equations. Since we've got a system of linear equations, we'll usu- the system is determined by its coefficients and its ally let m denote the number of equations, and n constants, and those can naturally be displayed in the number of variables, and say the system is a a matrix. m × n system of equations. For instance, here's an Consider the 3×2 system of equations above. Its example 3 × 2 system of equations: coefficients can be displayed in the 3 × 2 matrix 2 5 2 3 8 3 −1 < 5x + 2y = 12 4 5 3x − y = 5 1 3 : x + 3y = 5 called the coefficient matrix for the system. If you want to include the constants a a matrix, too, you Here, m = 3 since there are 3 equations, and n = 2 can add another column to get the 3 × 3 matrix since the two variables are x and y. (Typically, 2 5 2 12 3 when a system has more equations than unknowns, 3 −1 5 then there are no solutions. Does this system have 4 5 1 3 5 any solutions?) When there are many variables, we usually sub- called the augmented matrix for the system. Fre- script them, but when there are few variables we quently, a vertical line is added to separate the co- usually use different letters for the variables. For efficients from the constants, as in instance, if n = 6, we might use the 6 variables 2 5 2 12 3 x1; x2; x3; x4; x5, and x6, but if n = 3, we might use 4 3 −1 5 5 the three variables x, y, and z. 1 3 5 1 We'll only draw the vertical line to emphasize the meanings of the different parts of the matrix. We'll 2.0000 usually omit it. 1.0000 0 Matrices in Matlab. We've seen how easy it is to enter into Matlab the coefficient matrix A Thus, (x; y; z) = (2; 1; 0) is a solution to the sys- and constant column vector b of a system of linear tem of equations. But it's not the only solution. equations, and then compute the solution. Let's see Another is (x; y; z) = (3; −7; 11), and there are in- what Matlab does for the 3×2 system of equations finitely more solutions to this indeterminate sys- above. tem. On the other hand, the system >> A = [5 2; 3 -1; 1 3] A = 3x + 2y = 4 5 2 6x + 4y = 5 3 -1 is inconstant, that is, it has no solutions. Matlab 1 3 gives >> b = [12; 5; 5] >> A=[3 2;6 4] b = A = 12 3 2 5 6 4 5 >> b=[4;5] >> A\b b = ans = 4 2.0000 5 1.0000 >> A\b Let's try on this system Matlab Warning: Matrix is singular. 5x + 2y + z = 12 3x − y − z = 5 ans = -Inf >> A=[5 2 1;3 -1 -1] Inf A = 5 2 1 We'll see later how Matlab can be used to help 3 -1 -1 you find all the solutions to a system. Matlab also allows you to do create the iden- >> b=[12;5] tity matrices of various sizes, do addition and sub- b = traction on matrices of the same shape, multiply a 12 scalar and a matrix, take a transpose of a matrix, 5 as well as many other operations, most of which we'll use throughout the course. >> A\b ans = >> I = eye(3) 2 I = system of linear equations. The simplest ones are 1 0 0 called elementary row operations, and the elemen- 0 1 0 tary ones are of three types. 0 0 1 1. Exchange two rows. >> A=[4 5; 2 3; 0 1] 2. Multiply or divide a row by a nonzero constant. A = 4 5 3. Add or subtract a multiple of one row from 2 3 another. 0 1 You can use these three operations to convert the >> B=[1 1; 3 0; 4 4] augmented matrix to a particularly simple form B = that allows you to read off the solutions. The idea 1 1 is to eliminate the variables from the equations so 3 0 that each variable only occurs once. If there's a 4 4 unique solution, that works perfectly. If the system is indeterminate and has infinitely many solutions, >> A+B it doesn't work perfectly, but we'll see how that ans = works. 5 6 You can systematize this elimination process to 5 3 form an algorithm. Work from the leftmost column 4 5 right one column at a time to simplify the aug- mented matrix (and, therefore, the system it repre- >> 2*A-B sents). The column you're working on is called the ans = pivot column. Look down the column for the first 7 9 nonzero entry that has all 0s to its left. Call that 1 6 entry the pivot element, or more simply the pivot. -4 -2 There are three things to do corresponding to the three elementary row operations. >> B' 1. Exchange the pivot row with the highest row ans = that has 0s in the pivot column and all columns 1 3 4 to the left. (Do nothing if there is no such row.) 1 0 4 2. Divide the pivot row by the pivot so that the Elementary row operations, and reduced row value of the pivot becomes 1. echelon form for a matrix. When you want to 3. Subtract multiples of the pivot row from the solve a system of linear equations Ax = b, form other rows to clear out the pivot column (ex- the augmented matrix by appending the column b cept the pivot itself). to the right of the coefficient matrix A. Then you can solve the system of equations by operating on When you're all done with this algorithm, the the rows of the augmented matrix rather than on matrix will be in something called reduced row ech- the actual equations in the system. The three row elon form. The word echelon comes from a partic- operations are those operations on a matrix that ular stairstep formation of troops. For us, a matrix don't change the solution set of the corresponding is in reduced row echelon form if 3 1. the rows of all zeros (if any) appear at the bot- because 0 = (0; 0; 0) is always a solution, if v is a tom of the matrix solution, then any scalar multiple of v is a solution, and if both v and w are solutions, then so is v +w. 2. the first nonzero entry of a nonzero row is 1 3. that leading 1 appears appears to the right of Row reduction in Matlab. You can actually leading 1s in higher rows perform individual elementary row operations in Matlab, either by explicitly manipulating the 4. all the other entries in a column that has a rows of a matrix or by using the routine reduce. leading 1 are 0 If you're only interested in the resulting reduced row eschelon form of a matrix, then you can use the If the first three conditions hold, the matrix is said command rref. For example, here it's used for the to be in row echelon form. system of linear equations The reduction algorithm to convert an aug- mented matrix to reduced row echelon form goes 8 < 2v + w − 3x + 4y + 5z = 7 by the name Gauss-Jordan reduction. The partial v − 2w + 2x + 3y = 4 algorithm that stops with a row echelon form goes : 4v + 5x + 2y + z = −4 by the name Gaussian elimination. As we've seen, this reduction was known to the ancient Chinese. >> A=[2 1 -3 4 5 7;1 -2 2 3 0 4;4 0 5 2 1 -4] Two matrices are row equivalent if their corre- A = sponding systems of linear equations have the same 2 1 -3 4 5 7 solutions. It's easy to show that means you can 1 -2 2 3 0 4 perform a sequence of elementary row operations 4 0 5 2 1 -4 on one to eventually get the other.
Load more

Recommended publications
  • Formal Power Series - Wikipedia, the Free Encyclopedia
    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
    [Show full text]
  • 5. Polynomial Rings Let R Be a Commutative Ring. a Polynomial of Degree N in an Indeterminate (Or Variable) X with Coefficients
    5. polynomial rings Let R be a commutative ring. A polynomial of degree n in an indeterminate (or variable) x with coefficients in R is an expression of the form f = f(x)=a + a x + + a xn, 0 1 ··· n where a0, ,an R and an =0.Wesaythata0, ,an are the coefficients of f and n ··· ∈ ̸ ··· that anx is the highest degree term of f. A polynomial is determined by its coeffiecients. m i n i Two polynomials f = i=0 aix and g = i=1 bix are equal if m = n and ai = bi for i =0, 1, ,n. ! ! Degree··· of a polynomial is a non-negative integer. The polynomials of degree zero are just the elements of R, these are called constant polynomials, or simply constants. Let R[x] denote the set of all polynomials in the variable x with coefficients in R. The addition and 2 multiplication of polynomials are defined in the usual manner: If f(x)=a0 + a1x + a2x + a x3 + and g(x)=b + b x + b x2 + b x3 + are two elements of R[x], then their sum is 3 ··· 0 1 2 3 ··· f(x)+g(x)=(a + b )+(a + b )x +(a + b )x2 +(a + b )x3 + 0 0 1 1 2 2 3 3 ··· and their product is defined by f(x) g(x)=a b +(a b + a b )x +(a b + a b + a b )x2 +(a b + a b + a b + a b )x3 + · 0 0 0 1 1 0 0 2 1 1 2 0 0 3 1 2 2 1 3 0 ··· Let 1 denote the constant polynomial 1 and 0 denote the constant polynomial zero.
    [Show full text]
  • Formal Power Series Rings, Inverse Limits, and I-Adic Completions of Rings
    Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely many variables over a ring R, and show that it is Noetherian when R is. But we begin with a definition in much greater generality. Let S be a commutative semigroup (which will have identity 1S = 1) written multi- plicatively. The semigroup ring of S with coefficients in R may be thought of as the free R-module with basis S, with multiplication defined by the rule h k X X 0 0 X X 0 ( risi)( rjsj) = ( rirj)s: i=1 j=1 s2S 0 sisj =s We next want to construct a much larger ring in which infinite sums of multiples of elements of S are allowed. In order to insure that multiplication is well-defined, from now on we assume that S has the following additional property: (#) For all s 2 S, f(s1; s2) 2 S × S : s1s2 = sg is finite. Thus, each element of S has only finitely many factorizations as a product of two k1 kn elements. For example, we may take S to be the set of all monomials fx1 ··· xn : n (k1; : : : ; kn) 2 N g in n variables. For this chocie of S, the usual semigroup ring R[S] may be identified with the polynomial ring R[x1; : : : ; xn] in n indeterminates over R. We next construct a formal semigroup ring denoted R[[S]]: we may think of this ring formally as consisting of all functions from S to R, but we shall indicate elements of the P ring notationally as (possibly infinite) formal sums s2S rss, where the function corre- sponding to this formal sum maps s to rs for all s 2 S.
    [Show full text]
  • The Method of Coalgebra: Exercises in Coinduction
    The Method of Coalgebra: exercises in coinduction Jan Rutten CWI & RU [email protected] Draft d.d. 8 July 2018 (comments are welcome) 2 Draft d.d. 8 July 2018 Contents 1 Introduction 7 1.1 The method of coalgebra . .7 1.2 History, roughly and briefly . .7 1.3 Exercises in coinduction . .8 1.4 Enhanced coinduction: algebra and coalgebra combined . .8 1.5 Universal coalgebra . .8 1.6 How to read this book . .8 1.7 Acknowledgements . .9 2 Categories { where coalgebra comes from 11 2.1 The basic definitions . 11 2.2 Category theory in slogans . 12 2.3 Discussion . 16 3 Algebras and coalgebras 19 3.1 Algebras . 19 3.2 Coalgebras . 21 3.3 Discussion . 23 4 Induction and coinduction 25 4.1 Inductive and coinductive definitions . 25 4.2 Proofs by induction and coinduction . 28 4.3 Discussion . 32 5 The method of coalgebra 33 5.1 Basic types of coalgebras . 34 5.2 Coalgebras, systems, automata ::: ....................... 34 6 Dynamical systems 37 6.1 Homomorphisms of dynamical systems . 38 6.2 On the behaviour of dynamical systems . 41 6.3 Discussion . 44 3 4 Draft d.d. 8 July 2018 7 Stream systems 45 7.1 Homomorphisms and bisimulations of stream systems . 46 7.2 The final system of streams . 52 7.3 Defining streams by coinduction . 54 7.4 Coinduction: the bisimulation proof method . 59 7.5 Moessner's Theorem . 66 7.6 The heart of the matter: circularity . 72 7.7 Discussion . 76 8 Deterministic automata 77 8.1 Basic definitions . 78 8.2 Homomorphisms and bisimulations of automata .
    [Show full text]
  • NEW TRENDS in SYZYGIES: 18W5133
    NEW TRENDS IN SYZYGIES: 18w5133 Giulio Caviglia (Purdue University, Jason McCullough (Iowa State University) June 24, 2018 – June 29, 2018 1 Overview of the Field Since the pioneering work of David Hilbert in the 1890s, syzygies have been a central area of research in commutative algebra. The idea is to approximate arbitrary modules by free modules. Let R be a Noetherian, commutative ring and let M be a finitely generated R-module. A free resolution is an exact sequence of free ∼ R-modules ···! Fi+1 ! Fi !··· F0 such that H0(F•) = M. Elements of Fi are called ith syzygies of M. When R is local or graded, we can choose a minimal free resolution of M, meaning that a basis for Fi is mapped onto a minimal set of generators of Ker(Fi−1 ! Fi−2). In this case, we get uniqueness of minimal free resolutions up to isomorphism of complexes. Therefore, invariants defined by minimal free resolutions yield invariants of the module being resolved. In most cases, minimal resolutions are infinite, but over regular rings, like polynomial and power series rings, all finitely generated modules have finite minimal free resolutions. Beyond the study of invariants of modules, syzygies have connections to algebraic geometry, algebraic topology, and combinatorics. Statements like the Total Rank Conjecture connect algebraic topology with free resolutions. Bounds on Castelnuovo-Mumford regularity and projective dimension, as with the Eisenbud- Goto Conjecture and Stillman’s Conjecture, have implications for the computational complexity of Grobner¨ bases and computational algebra systems. Green’s Conjecture provides a link between graded free resolutions and the geometry of canonical curves.
    [Show full text]
  • Coalgebras from Formulas
    Coalgebras from Formulas Serban Raianu California State University Dominguez Hills Department of Mathematics 1000 E Victoria St Carson, CA 90747 e-mail:[email protected] Abstract Nichols and Sweedler showed in [5] that generic formulas for sums may be used for producing examples of coalgebras. We adopt a slightly different point of view, and show that the reason why all these constructions work is the presence of certain representative functions on some (semi)group. In particular, the indeterminate in a polynomial ring is a primitive element because the identity function is representative. Introduction The title of this note is borrowed from the title of the second section of [5]. There it is explained how each generic addition formula naturally gives a formula for the action of the comultiplication in a coalgebra. Among the examples chosen in [5], this situation is probably best illus- trated by the following two: Let C be a k-space with basis {s, c}. We define ∆ : C −→ C ⊗ C and ε : C −→ k by ∆(s) = s ⊗ c + c ⊗ s ∆(c) = c ⊗ c − s ⊗ s ε(s) = 0 ε(c) = 1. 1 Then (C, ∆, ε) is a coalgebra called the trigonometric coalgebra. Now let H be a k-vector space with basis {cm | m ∈ N}. Then H is a coalgebra with comultiplication ∆ and counit ε defined by X ∆(cm) = ci ⊗ cm−i, ε(cm) = δ0,m. i=0,m This coalgebra is called the divided power coalgebra. Identifying the “formulas” in the above examples is not hard: the for- mulas for sin and cos applied to a sum in the first example, and the binomial formula in the second one.
    [Show full text]
  • Math 476 - Abstract Algebra 1 Day 33 Group Assignment Name
    Math 476 - Abstract Algebra 1 Day 33 Group Assignment Name: Polynomial Rings n n−1 Definition 11.2 Let R be a commutative ring. A polynomial in x over R is an expression of the form: anx +an−1x + 2 1 0 ··· + a2x + a1x + a0x , where n is a non-negative integer, and an,an−1, ··· ,a2,a1,a0 are elements of R. The set of all polynomials over the ring R will be denoted by R[x]. We will generally assume that an =6 0. • The symbol x is called an indeterminate. It is to be regarded as a formal symbol and not as an element of the ring 0 1 2 n R. In effect, the symbols x ,x ,x , ··· ,x serve as placeholders alongside the ring elements a0,a1, ··· ,an. k • The expressions akx are called the terms of the polynomial. The elements a0,a1, ··· ,an in the ring R are called k the coefficients of the polynomial p(x). We call ak the coefficient of x in the representation of p(x). • When working with a polynomial, instead of writing x1, we simply write x. In addition, we usually do not write x0 and 0 we will write a0x simply as a0 (does this make sense when R does not have an element 1R?). Using these conventions, n n−1 2 we can write p(x) in the form p(x) = anx + an−1x + ··· + a2x + a1x + a0, where n is a non-negative integer, an,an−1, ··· ,a2,a1,a0 are elements of R, and an =6 0.
    [Show full text]
  • BOUNDS and DEFINABILITY in POLYNOMIAL RINGS Introduction the Main Theme of This Article Is the Existence of Bounds for Basic
    BOUNDS AND DEFINABILITY IN POLYNOMIAL RINGS MATTHIAS ASCHENBRENNER Abstract. We study questions around the existence of bounds and the de- pendence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor. In particular, we show that the module of syzy- gies of polynomials f1, . , fn ∈ R[X1,...,XN ] with coefficients in a Pr¨ufer domain R can be generated by elements whose degrees are bounded by a num- ber only depending on N, n and the degree of the fj . This implies that if R is a B´ezout domain, then the generators can be parametrized in terms of the coefficients of f1, . , fn using the ring operations and a certain division function, uniformly in R. Introduction The main theme of this article is the existence of bounds for basic operations of linear algebra in polynomial rings over (commutative) rings of an arithmetic nature. The following result, shown in Section 3 below, is typical. Theorem A. Given integers N, d, n > 0 there exists an integer β = β(N, d, n) with the following property: for every Pr¨ufer domain R and polynomials f1, . , fn ∈ n R[X] = R[X1,...,XN ] of (total) degree 6 d, the R[X]-submodule of R[X] con- sisting of all solutions to the linear homogeneous equation f1y1 + ··· + fnyn = 0 can be generated by (finitely many) solutions whose components have degree 6 β. A classical theorem due to G. Hermann [24] states that Theorem A is true if we replace “Pr¨ufer domain” by “field”. In this case, it is easy to see that β can be chosen independent of n; Seidenberg [36] computed an explicit (doubly exponential) bound β.
    [Show full text]
  • Vector Spaces
    Chapter 2 Vector Spaces 2.1 INTRODUCTION In various practical and theoretical problems, we come across a set V whose elements may be vectors in two or three dimensions or real-valued functions, which can be added and multi- plied by a constant (number) in a natural way, the result being again an element of V. Such concrete situations suggest the concept of a vector space. Vector spaces play an important role in many branches of mathematics and physics. In analysis infinite dimensional vector spaces (in fact, normed vector spaces) are more important than finite dimensional vector spaces while in linear algebra finite dimensional vector spaces are used, because they are simple and linear transformations on them can be represented by matrices. This chapter mainly deals with finite dimensional vector spaces. 2.2 DEFINITION AND EXAMPLES VECTOR SPACE. An algebraic structure (V, F, ⊕, ) consisting of a non-void set V , a field F, a binary operation ⊕ on V and an external mapping : F ×V → V associating each a ∈ F,v ∈ V to a unique element a v ∈ V is said to be a vector space over field F, if the following axioms are satisfied: V -1 (V, ⊕) is an abelian group. V -2 For all u, v ∈ V and a, b ∈ F, we have (i) a (u ⊕ v)=a u ⊕ a v, (ii) (a + b) u = a u ⊕ b u, (iii) (ab) u = a (b u), (iv) 1 u = u. The elements of V are called vectors and those of F are called scalars. The mapping is called scalar multiplication and the binary operation ⊕ is termed vector addition.
    [Show full text]
  • 1 Linear Algebra
    Mathematics 595 (CAP/TRA) Introduction to Linear Analysis Fall 2005 1 Linear Algebra 1.1 Vector spaces Definition 1.1.1. Let F be a field. A vector space over F is a set V of elements, called vectors, equipped with two operations + : V × V ! V (addition) and · : F × V ! V (scalar multiplication) such that (V; +) is an abelian group, and the following compatibility relations hold: 1. (distributivity) a; b 2 F, v; w 2 V ) a(v + w) = av + aw, (a + b)v = av + bv, 2. (associativity of multiplication) a(bv) = (ab)v, 3. (identity) 1v = v. Here 1 denotes the unit element of F. Notice that we typically omit the notation · for scalar multiplication, writing av = a · v. Examples 1.1.2. 1. n-dimensional Euclidean space Rn is a vector space over R. 2. The space C[0; 1] of continuous functions on the interval [0; 1] is a vector space over R. 3. Let k = 0; 1; 2; : : :. The space Ck[0; 1] of functions f on the interval [0; 1] whose kth derivative f (k) exists and is continuous is a vector space over R. Similarly, the space C1[0; 1] of functions on [0; 1] for which f (k) exists for all k, is a vector space over R. 4. R is a vector space over Q (see Exercise 1.1.17). 5. This example requires some basic fluency in abstract algebra. Let p be a prime and let K be a finite field of characteristic p. Then K is a vector space over Zp.
    [Show full text]
  • Prime Ideals in Polynomial Rings in Several Indeterminates
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 67–74 S 0002-9939(97)03663-0 PRIME IDEALS IN POLYNOMIAL RINGS IN SEVERAL INDETERMINATES MIGUEL FERRERO (Communicated by Ken Goodearl) Abstract. If P is a prime ideal of a polynomial ring K[x], where K is a field, then P is determined by an irreducible polynomial in K[x]. The purpose of this paper is to show that any prime ideal of a polynomial ring in n-indeterminates over a not necessarily commutative ring R is determined by its intersection with R plus n polynomials. Introduction Let K be a field and K[x] the polynomial ring over K in an indeterminate x.If Pis a prime ideal of K[x], then there exists an irreducible polynomial f in K[x] such that P = K[x]f. This result is quite old and basic; however no corresponding result seems to be known for a polynomial ring in n indeterminates x1, ..., xn over K. Actually, it seems to be very difficult to find some system of generators for a prime ideal of K[x1, ..., xn]. Now, K[x1, ..., xn] is a Noetherian ring and by a converse of the principal ideal theorem for every prime ideal P of K[x1, ..., xn] there exist n polynomials f1, ..., fn such that P is minimal over (f1, ..., fn), the ideal generated by f1, ..., fn ([4], Theorem 153). Also, as a consequence of ([1], Theorem 1) it follows{ that any} prime ideal of K[x1, ..., xn] is determined by n polynomials.
    [Show full text]
  • Arxiv:0801.0483V1 [Math.AC] 3 Jan 2008 N,Rsetvl,Dfie Sfollows: As Defined Respectively, And, Oue R Unital
    GLOBAL GORENSTEIN DIMENSIONS OF POLYNOMIAL RINGS AND OF DIRECT PRODUCTS OF RINGS DRISS BENNIS AND NAJIB MAHDOU Abstract. In this paper, we extend the well-known Hilbert’s syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Goren- stein homological dimensions of direct products of rings. Our results generate examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension. 1. Introduction Throughout this paper all rings are commutative with identity element and all modules are unital. Setup and Notation: Let R be a ring, and let M be an R-module. We use pdR(M), idR(M), and fdR(M) to denote, respectively, the classical projec- tive, injective, and flat dimensions of M. We use gldim(R) and wdim(R) to denote, respectively, the classical global and weak dimensions of R. It is by now a well-established fact that even if R to be non-Noetherian, there exist Gorenstein projective, injective, and flat dimensions of M, which are usually denoted by GpdR(M), GidR(M), and GfdR(M), respectively. Some references are [1, 2, 4, 5, 7]. Recently, the authors in [2] started the study of global Gorenstein dimensions of rings, which are called, for a ring R, Gorenstein projective, injective, and weak dimensions of R, denoted by GPD(R), GID(R), and G−wdim(R), respectively, and, respectively, defined as follows: GPD(R) = sup{GpdR(M) | M R−module}, arXiv:0801.0483v1 [math.AC] 3 Jan 2008 GID(R) = sup{GidR(M) | M R−module}, and G−wdim(R) = sup{GfdR(M) | M R−module}.
    [Show full text]