Azumaya algebras on fields

MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS DANG THI NGOC THANH AZUMAYA ALGEBRAS ON FIELDS BACHELOR THESIS Hanoi - 2019 MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS DANG THI NGOC THANH AZUMAYA ALGEBRAS ON FIELDS BACHELOR THESIS SUPERVIOR: PHAM THANH TAM Hanoi - 2019 Acknowledgement Firstly, I would like to express my sincere gratitude to my advisor MCs Pham Thanh Tam for sharing expertise, valuable guidance and encouragement extended to me I am also grateful to the Department faculty members in the geometry group of Hanoi Pedagogical University No for their insightful comments and stimulation, but also for the hard question which incented me to widen my studying from various perspectives And finally, my deepest thank is to my family who always shows their concern about me and supports me to finish this paper Due to time, capacity and conditions are limited, so that my thesis cannot avoid mistakes Therefore, I am looking forward to receiving comments and recommendations from teachers and friends Hanoi, May, 2019 Student Dang Thi Ngoc Thanh Assurance I hereby declare that this thesis is my own work and to the best of my knowledge, it contains no material previously published or written by another person, nor material which to a substantial extent has been accepted for award of any other degree or diploma at any educational institution, except where due acknowledgment is made in the thesis I also assure that all the help for this thesis has been acknowledged and the results presented in the thesis have been identified clearly Hanoi, May, 2019 Student Dang Thi Ngoc Thanh Contents Preliminaries 1.1 Matrix algebras 1.2 Semi - simplicity 1.3 Tensor products 1.4 Central simple algebras Azumaya algebras over fields 14 2.1 Azumaya algebras 14 2.2 Azumaya algebras over fields 16 2.3 Examples of Brauer groups 20 Bibliography 23 Introduction Mathematics is an important field and serves so many of the sciences that it is a prerequisite for studying every scientific discipline The matrix algebras over a field have many especially application in studying the structure of the linear automorphism of finite dimensional vector spaces The Azumaya algebras on fields is a generalization of center simple algebras In particular, it is a generalization of matrix algebras on a field Such a notion was introduced in [GA51] for the case of a field The studying of Azumaya algebras on a field is really significant in extending the research of central simple algebras and matrix algebras An Azumaya algebra over a field R is an R-algebras A that is free and of finite rank r ❈ as an R-module, such that the tensor product (where A❳ is the opposite algebra) is isomorphic to the matrix algebra EndR ❼A➁ ☎ Mr ❼R➁ Two Azumaya algebras A1 and A2 over R are similar if there exist locally free R-modules E1 and E2 of finite positive rank such that A1 ❛ End❼E1 ➁ ✟ A2 ❛ End❼E2 ➁, where End❼Ei ➁ is the endomorphism sheaf of Ej The set of similar classes of Azumaya algebras with operation given by tensor product has form a group This group is called Brauer group of field R Base on the basic knowledge about algebraic geometry and desiring comprehensive improvement of Mathematics,the researcher has selected a topic “Azumaya algebras on fields” for Bachelor Thesis This Thesis is organized as follows In Chapter 1, I will introduce algebras over an arbitrary commutative ring and some examples of algebras Namely matrix algebras, semisimplicity, and tensor products of algebras will be presented in this chapter Additionally, a brief explanation of the theory of semi-simple rings and modules, and its connection with central simple algebras is given In Chapter 2, we begin by Azumaya algebras and Brauer group on fields Moreover, examples of Brauer groups are given, including the Brauer group of a quasi-algebraically close field This Thesis is organized as follows Chapter Preliminaries Let R be a commutative ring and R-algebra, is a pair ❼A, ϕ➁ including a ring A and a ring homomorphism ϕ ✂ R   Z ❼A➁ called the structure map of A over R, where Z ❼A➁   ➌x ❃ A ✂ xa   ax, ➛a ❃ A➑ is call the center of A, which is a subring of A If A is an R-algebra that is a division ring, then we say that A is a division algebra 1.1 Matrix algebras Let R be a commutative ring and the additive group A✔ of R-algebra is a R- module by restriction of scalars, i.e., by the composed map R ϕ f i ÐÐÐÐ   Z ❼A➁ ÐÐР A ÐÐÐР End❼A✔ ➁ Where i is the inclusion map, f : the usual A-module structure of ❼A✔ ➁ Р End❼A➁ a ③  a A Р A m ③  am f ✂A ✂ Now, given A be any ring and M be A-module (i.e, ➛a ❃ A, m ❃ M then a.m ❃ M ➁ Then the endomorphism ring EndA ❼M ➁ is the ring consisting of A-linear endomorphisms of M ,where multiplication and addition are given by composition and pointwise addition, respectively Let λa denote the A-linear endomorphism of M given by left-multiplication by a ➛ ϕ✂m   m ❃ EndA❼m➁ Then we have λa ϕ❼m➁   a.ϕ❼m➁   ϕ❼am➁   ϕ❼λa ❼m➁➁   ϕλa ❼m➁ Thus, λa is clearly in the center of EndA ❼M ➁ if a ❃ Z ❼A➁ Consider the map Р Z ❼EndA❼M ➁➁ a ③  λa g ✂ Z ❼ A➁ is a ring homomorphism, which in particular gives EndA ❼M ➁✔ the usual Z ❼A➁✏ module structure Moreover, let be a commutative ring and a ring homomorphism h ✂ R R   Z ❼A➁ Then the composed map g h ÐÐÐÐÐÐÐ   Z ❼A➁ ÐÐÐÐÐÐÐ   Z ❼EndA❼M ➁➁ is called an R-algebra structure on EndA ❼M ➁ An algebra of this form is often referred to as an endomorphism algebra Definition 1.1 Let R be a commutative ring, the ring Mn ❼R➁ is the matrix ring on R Proposition 1.2 Let A be a ring, n ❃ Z❆0 So there is a bijection between the set of two-sided ideals A and the set of two-sided ideals of Mn ❼A➁ that sends an two-sided ideal I of A to the two-sided ideal Mn ❼I ➁ of Mn ❼A➁ Proof See [ Row88,Proposition 1.1.5] Corollary 1.3 Let A a simple ring, n ❃ Z❆0 Then Mn ❼A➁ is a simple ring Definition 1.4 Let R be a ring The opposite ring R❳ of R is the additive group R✔ endowed with a multiplication operation “❻” defined for x, y ❃ R✔ by x ❻ y   y   x , where “ ” is the multiplication operation of R Definition 1.5 Let R be a commutative ring and let ❼A, ϕ➁ be an algebra over R A-algebra ❼B, ψ ➁ is called the opposite algebra of A if B is A❳ as a ring, and ϕ is equal to ψ Definition 1.6 Let R be a commutative ring and A be an R-algebra Then there is an R-algebra isomorphism between A❳ and EndA ❼A A➁ that maps an element x in A❳ to the A-linear endomorphism of A given by right-multiplication with x Remark Let R be a commutative ring and A be an R-algebra Let M be a A-module n ❃ Z❆0 Then we obtain the R-algebra isomorphism EndA ❼M n ➁ ☞ Mn ❼EndA ❼M ➁➁ In particular, if M is a free A-module of rank n, that is, if M is isomorphic to An as an A-module, then there is an R-algebra isomorphism EndA ❼M ➁ ☞ Mn ❼A❳ ➁ by above proposition 1.2 Semi - simplicity Definition 1.7 Let R be a ring, M be a R-module If M has two Rsubmodule (0 and itself), then M is called simple Let k be a field, the additive group A✔ of a k-algebra A is a k-vector space The k-dimension that A✔ he is called hereby gets is called the rank of the algebra and denotes A ✂ k ✆ If  A ✂ k ✆ is finite, then we say that A has finite rank over k or that A is of finite rank over k Definition 1.20 Let k be a field, a central simple k-algebra A is a finite dimensional k-algebra with center k and non-trivial two-sided ideal Example 1.21 Every field is a central simple algebra over itself Example 1.22 Any division algebra over k is clearly a central simple algebra since any non-zero element is a unit For example, we get quaternion algebras H ❼a, b➁   spank ➌1, i, j, ij ➑ With multiplication given by i2   a, j   b, ij   ✏ij For example, when k   R, a   b   ✏1,we recover the familiar quaternion H Definition 1.23 Let k be a field, D be an algebra over k if D is a central algebra of finite rank over k that is also a division ring, so D is called a central division algebra over k Proposition 1.24 Let k be a field, A be a k-algebra So A is a simple algebra of finite rank over k if and only if there exist n ❃ Z❆0 and a division algebra D of finite rank over k such that A is isomorphic to M n ❼D➁ as k-algebra n ❃ R is uniquely determined, as is the division algebra D up to isomorphism Corollary 1.25 Let k be a field, A be a k-algebra So A is a central simple algebra of finite rank over k and only if there exist Z❆0 and a division algebra D of finite rank over k such that A is isomorphic M n ❼D➁ to as k-algebra n ❃ R is uniquely determined, as is the division algebra D up to isomorphism Proof This follows from the previous proposition and the obvious k-algebra isomorphisms k ☞ Z ❼A➁ ☞ Z ❼D➁ Theorem 1.26 (Wedderburn) Let A be a central simple algebra over k, so A ☞ M n ❼D➁ for some n ❈ and k-central division algebra D Proposition 1.27 Let k be a field, A be a central simple k-algebra, B be also a central simple k-algebra So A❛R B is a central simple k-algebra Proof See [Ker07, 2.6] Proposition 1.28 Let k be a field, A be a central simple k-algebra, then Aop ✂  ➌aop ❙a ❃ A, aop bop   ba➑ is a central simple algebra over k Furthermore, there is an isomorphism ✏ ÐÐÐÐ   Endk ❼A➁ b ③  ❼x ✭ axb➁ A ❛ k Aop a❛ Notice that the choice of a basis gives an isomorphism Endk ❼A➁ with a matrix algebra Lemma 1.29 Let k be an algebraically closed field, any division algebras of finite rank over k is isomorphic to k as an algebra 10 Proof Suppose D is a non-commutative division algebra of finite rank over k Let x be an element of D So, x generates the commutative extension of k, which is finite for D is of finite field extensions are algebraic, k is algebraically closed, it follows that k ❼x➁   k holds Consequently, x is an element of k, that is, we get that D   k holds Definition 1.30 Let a be a ring, X be a subset of A So C A ❼X ➁   ➌a ❃ A ✂ xa   ax for all ➌x ❃ X ➑ is called the centralizer of X in A It is easy to check that the centralizer of a subset a ring A which is a subring of A as well Theorem 1.31 Let A be a k-algebra, N a faithfully, semi-simple A module, hence C ❼C ❼A➁➁   Endk ❼N ➁ where the centralizers are taken in Endk ❼N ➁ Corollary 1.32 Every central simple k-algebra is isomorphism to for some division k-algebra D Proof Choose a simple A-module S ( for example, we choose a minimal left ideal of A) A acts faithfully on S where the Kernel of A   Endk ❼S ➁ is a two-sided ideal not containing Let D be a the centralizer of A in the k-algebra Endk ❼S ➁, by the theorem 1.21, we get A   C ❼D➁ , i.e., A   EndD ❼S ➁ But S is a simple 11 A-module, so for a ❃ D multiplication by a is an A-linear endomorphism a✂S Р S, then is either or veritable by Schur’s Lemma Since the inverse is also A-linear and D   C ❼A➁, it follows D is a division k-algebra, that is S ☞ Dn for some n Thus, A   EndD ❼Dn ➁   M n ❼Dopp ➁ Proposition 1.33 Let A be a central simple algebra over k, l ❃ k is a field extension Then A❛k ❼l➁ is a central simple algebra over l Proof Suppose that A is central simple k-algebra, then Al As  Al ✂ l✆    A ✂ l✆ holds, we see that Al is central simple algebra over l Conversely, suppose that A❛k ❼l➁ is central simple algebra over l, the k-algebra isomorphism Z ❼A➁ ❛k l ☞ l holds So we get the k-algebra isomorphism Z ❼A➁ ☞ k Suppose that A is not simple, let I be a two-sided ideal of A So I ❛k is two-sided ideal of Al , which is simple This contradiction implies that A is simple as a ring Proposition 1.34 If A is a central simple algebra over k, so  A ✂ k ✆ is a square Proof  A ✂ k ✆   ✁A ❛ k k¯ ✂ k¯✝ But A❛k k¯ is a central simple algebra over k¯ and then is isomorphism to M n ❽k¯➂ for some n Hence,  A ✂ k ✆   n2 12 Lemma 1.35 Let k be a field, A be a central simple algebra over k Let n be the degree of A So A❛k A❳ is isomorphic to Mn ❼k ➁ as a k-algebra 13 Chapter Azumaya algebras over fields 2.1 Azumaya algebras Definition 2.1 Let R be a commutative ring and let A be an associative R-algebra such as R   A identifies R with a subring of Z ❼A➁ We say that A is an Azumaya algebra over R if A is free of finite rank l as an R-module and if the following map Р EndR ❼A➁ a ③  ❽x ✭ axa ➂ φA ✂ A❛R Ao a❛ ➐ ➐ is an isomorphism Where Ao is the opposite algebra to A Example 2.2 Let A be a central simple algebra over a field F ,the ring D   Mn ❼F ➁   EndF ❼A➁ ia an Azumaya algebra over F Proposition 2.3 If A is a central simple algebra over a field k, so A is Azumaya over k Proof Let dimk ❼A➁   l so φA is a morphism between k-algebras which both have dimension l2 over k Then it suffices to check injectivity Note 14 that A❛k Aop is a central simple algebra over k then ker ❼φA ➁   or A❛k Aop As the second option is obviously false we get proven injectivity of φA Proposition 2.4 Let A be an Azumaya algebra over R, so Z ❼A➁   R and there is a bijection between the (two-sided) ideal of A and the ideals of R: ➌Ideals 1✏1 ✂  ➌Ideals ofR➑ ■ ③  ■ R ❏ A ✂Ð❬ ❏ ofA➑ ✾ Proof Let ψ ❃ EndR ❼A➁ and c ❃ Z ❼A➁, so for all a ❃ A, we get cψ ❼a➁   ψ ❼ac➁   ψ ❼a➁ c Because ψ is given by multiplication by elements in A as A is Azumaya Similarly, ψ ❼I ➁ ❵ J for each ideal I of A Let   a1 , al be a basis for A as an R-module and define χi ❃ EndR ❼A➁ by χi ❼aj ➁   δij Write c   Pi riai with all ri ❃ R, so ◗ riai➅   ri ❃ R c   1.c   χ1 ❼a1 ➁ c   χ1 ❼a1 c➁   χ1 ❼1.c➁   χ1 ➀ l i 1 Now, we check the bijection between the sets of ideals As the maps are well defined, it suffices to prove I   ❼I ✾ R➁ A and J   JA ✽ R Both equalities are trivial to check Corollary 2.5 An Azumaya algebra over a field is a central simple algebra 15 2.2 Azumaya algebras over fields Let k be a field and CSA ❼k ➁ be the class of all central simple algebras over k We say that two central simple k-algebra A and B are similar, denoted by, if there are positive integers m and n such that Mm ❼A➁ is isomorphic to Mn ❼B ➁ as a k-algebra In the next lemma we prove that this defines an equivalence relation on CSA ❼k ➁ reducing to k-algebra isomorphism when the two central simple algebras have the same rank over k Lemma 2.6 Let k be a field So ✂ is an equivalence relation on CSA ❼k ➁, which reduces to k-algebra isomorphism when two central simple algebras have the same rank over k Proof We see that ✂ is clearly reflexive and symmetric on CSA ❼k ➁ Let A, B, and C be elements of CSA ❼k ➁ such that A ✂ B and B ✂ C So there are m, n, s, t ❃ Z❆0 such that the k -algebra isomorphisms Mm ❼A➁ ☞ Mn ❼B ➁ and Ms ❼B ➁ ☞ Mt ❼C ➁ hold It follows that the k-algebra isomorphisms Msm ❼A➁ ☞ Ms ❼Mm ❼A➁➁ ☞ Ms ❼Mn ❼B ➁➁ ☞ Mn ❼Ms ❼B ➁➁ ☞ Mn ❼Mt ❼C ➁➁ ☞ Mnt ❼C ➁ hold, so A ✂ C holds Consequently, ✂ is a transitive relation on CSA ❼k ➁, and thus an equivalence relation on CSA ❼k ➁ Suppose A, B ❃ CSA ❼k ➁ of the same rank that are m, n ❃ Z❆0 similar and so, there are such that the k-algebra isomorphism Mm ❼A➁ ☞ Mn ❼B ➁ We see that  Mm ❼A➁ ✂ k ✆   m2  A ✂ k ✆    Mn ❼B ➁ ✂ k ✆   n2  B ✂ k ✆ holds From which the equality  A ✂ k ✆    B ✂ k ✆ implies that m2   n2 holds 16 It is clear that k-algebra isomorphism Mm ❼A➁ ☞ Mm ❼B ➁ hold if and only if the k-algebra isomorphism A ☞ B holds Lemma 2.7 Let k be a field, let A, B, A➐ and B ➐ are central simple kalgebras such that A ✂ A➐ and B ✂ B ➐ Then A❛k B ✂ A➐ ❛k B ➐ Proof There exist m, n, s, t ❃ Z❆0 such that the k-algebra isomorphism Mm ❼A➁ ☞ Mn ❼A➐ ➁, Ms ❼B ➁ ☞ Mt ❼B ➐ ➁ hold.We get the k-algebra isomorphism Mm ❼A➁ ❛k Ms ❼B ➁ ☞ Mn ❼A➐ ➁ ❛k Mt ❼B ➐ ➁ And that Corollary 1.19 implies that we get the k-algebra isomorphism Mms ❼A❛k B ➁ ☞ Mnt ❼A➐ ❛k B ➐ ➁ Hence, A❛k B and A➐ ❛k B ➐ are similar Theorem 2.8 Let k be a field So there exist a pair ❼G, s➁ consisting of a group G and a surjective map s ✂ CSA ❼k ➁   G that for every central simple k-algebras A and B satisfies: i The equality s ❼A❛k B ➁   s ❼A➁ s ❼B ➁ holds ii The equality s ❼A➁   s ❼B ➁ holds if and only if A and B are similar Moreover, the pair ❼G, s➁ is uniquely determined up to a unique group isomorphism, that is, if ❼G➐ , s➐ ➁ is another pair satisfying the above, so there is a unique group isomorphism γ ✂ G   G➐ such that we get the equality s➐   γ ❳ s Proof Let H be a subclass of CSA ❼k ➁, this is a set such that every element of CSA ❼k ➁ is isomorphic as a k-algebra to at least one element of H Let 17 G be the quotient set of H by ✂ It is easy to show that a set dose exist For an element A of CSA ❼k ➁ ,let  A✆ denote the element of G that contains the elements of H that are similar to A, which gives a surjective map: Р G B ③   B ✆ s ✂ CSA ❼A➁ We show that G is an abelian group under the tensor product over k to the end, we see that the map Р G ❼ A✆ ,  B ✆➁ ③   A G✕G ❛ kB✆ Is well-defined by Lemma 2.7; So it remains to prove that G satisfies the group axioms and commutativity with respect to the tensor product We see that for any central simple k-algebra A it clearly holds that A❛k k is isomorphic to A as a k-algebra; then,  k ✆ functions as the identity element of G under the tensor product over k Associativity follows from Proposition 1.15 and commutativity follows from Proposition 1.14 At last the existence of inverse elements in G is proven by Lemma 1.35, which states that the inverse of an element  A✆ of G is given by the element containing the opposite algebra of A Hence, we get showed that G is an abelian group under the tensor product over k Furthermore, it is clear that for every A, B ❃ CSA ❼k ➁ the map s satisfies the equality s ❼A❛k B ➁   s ❼A➁ s ❼B ➁,then we get a pair ❼G, s➁ that satisfies the theorem Next, suppose that ❼G➐ , s➐ ➁ is another pair satisfies the theorem, and 18 the map Р G➐ ➐  A✆ ③  s ❼A➁ γ✂G It is easily checked that γ is a unique group isomorphism and so we have s➐   γ ❳ s It follows that ❼G, s➁ is uniquely determined up to isomorphism Definition 2.9 Let k be a field The group of the uniquely determined pair ❼G, s➁ is called the Brauer group over k, denoted by Br ❼k ➁ and is written multiplicatively For central simple algebra A over k, denote s ❼A➁ by  A✆ Moreover, an element b of is often denoted by  A✆, in which A is an element of CSA ❼k ➁ that is similar to an element b Proposition 2.10 Let k be a field Then every element of Br ❼k ➁ contains a unique central division k-algebra up to isomorphism Proof Let  A✆ be an element of Br❼k ➁ By Corollary 1.25 there exist a unique m, n ❃ Z❆0 and a unique central division k-algebra D up to isomorphism such that A is isomorphic to Mn ❼D➁ As a k-algebra It follows that D and A are similar; thus, D is an elements of  A✆ Suppose that D is another central division algebra over k that is similar to A, so there are m, n ❃ Z❆0 such that the k-algebra isomorphism Mm ❼D➁ ☞ Mn ❼D➐ ➁ holds Applying Corollary 1.25 to Mm ❼D➁ ☞ Mn ❼D➐ ➁, it immediately follows holds and that D is isomorphic to as a k-algebra 19 2.3 Examples of Brauer groups In the section, we will treat the following examples a Let k be an algebraically closed field, then Br ❼k ➁   b Finite fields have trivial Brauer group c Function fields of curses over algebraically closed fields have trivial Brauer group d Br❼R➁ ☞ Z⑦2Z Definition 2.11 Let i ❃ Z❆0 We say that a field k is Ci if for every n ❃ Z❆0 every non-constant homogeneous polynomial f ❃ k  X1 , Xn ✆ with i ❼deg f ➁ ❅ n has a non-trivial zero In particular, we say that k is quasi- algebraically if k is C1 Remark 2.12 It is clear that if a field is Ci for some i ❃ Z❆0 , so it is also for all j ❃ Z❈i Example 2.13 ❼ Being a C0 -field is equivalent to being algebraically closed ❼ For any i ❈ the field is not Ci ; the example x21 ✔ ✔ x2n for n ❈ shows this Theorem 2.14 (Chevalley-Warning) Let k be a field, m, n ❃ Z❆0 Let f1 , fm ❃ k ❼X1 , Xn ➁ be non-constant polynomials with P deg ❼fi➁ ❅ n i 1 m So the zero-locus Z ❼f1 , fm ➁ ❵ k n has cardinality divisible by p 20 Proof See [Ax64] Proposition 2.15 The Brauer group of a field is trivial Proof Let k be a C1 field For any finite-dimensional central division algebra D over k Let d2 be its dimension Now, we will construct a polynomial the violates the definition of a C1 field unless d ✏ holds and hence D ✏ k A reduced norm form associated to D is a homogeneous polynomial in d2 variables and has degree d Because the reduced norm restricts to a homomorphism D✕   k✕ and D is a division algebra, the norm form has no non-trivial zeroes This is contradicts k being a C1 field unless d   holds and so, the only possibility for D is k itself and hence Br ❼k ➁ is trivial Proposition 2.16 The Brauer group of a finite field is trivial By the Chevalley-Warning theorem, any finite field is C1 and then has trivial Brauer group by Proposition 2.15 21 Conclusion In this thesis, I have presented systematically the following result: Chapter 1: In this chapter, I present the theory of tensor products, semisimplicity, and matrix algebras Especially the definition, theorems and propositions of central simple algebras on a commutative ring Chapter 2: I mainly present Azumaya algebras over fields Theorem 2.8 for the structure of Azumaya algebras on a field that is the difference an equivalent relationship Moreover, I also present a very important result in the proposition 2.15 that Azumaya algebras over fields have the form of matrix algebra 22 Bibliography [Ax64] J.Ax, Zeroes of polynomials over finite fields, American Journal of Mathematics 86 (1964), no 2, 255-261 [Bou73] N.Bourbaki, Elements of mathematics-algebra, Springer-Verlag, 1973 [FD93] B Fard and K R Dennis, Noncommutative algebra, SpringerVerlag, 1993 [GA51] Goro Azumaya, On maximally central algebras, Nagoya Mathematical Journal ,1951 [Ja11] Abtien Javanpeykar, The Brauer group of a field, Bachelor’s thesis, 2011 [Ker07] I Kersten, Brauergruppen, Uniiversitatsverlag Gottingen, 2007 [Row88] L.Rowen, Ring theory volume 1, Academic Press, 1988 [Ve04] Vesselin Drensky, Edward Formanek (auth.),Polynomial Identity Rings, 2004 23 ... Central simple algebras Azumaya algebras over fields 14 2.1 Azumaya algebras 14 2.2 Azumaya algebras over fields 16 2.3 Examples... especially application in studying the structure of the linear automorphism of finite dimensional vector spaces The Azumaya algebras on fields is a generalization of center simple algebras In particular,... semisimplicity, and matrix algebras Especially the definition, theorems and propositions of central simple algebras on a commutative ring Chapter 2: I mainly present Azumaya algebras over fields Theorem