RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered optional for the course. However, although I have only highlighted the most important results here, you are responsible for all results stated in class Material is listed in the order it was covered in class. This mostly (but not entirely) corresponds to the order in which it appears in the book. 1. Chapter 11: Definitions and results Sections covered. All of Chapter 11 except 11.6 and 11.9. (We also include some material from 13.5 here, since it was done in that order in class) Definitions. Definition. A ring R is a set with two binary operations +, called addition and multiplication, that satisfy: (1) R is an abelian group under the operation +. (2) Multiplication is commutative and associative, and has an identity denoted 1. (3) For all a, b, c, we have (a + b)c = ac + bc. Definition. An element a R is a unit if it has a multiplicative inverse. Definition. If R is a ring, a subring of R is a subset which is closed under addition, multiplication, and subtraction, and which contains 1. Definition. Let R be a ring. Given a formal symbol x, a polynomial in x with coefficients in R is a finite formal sum f(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0, where each a i R. The a i are called the coefficients of f(x). The polynomial ring R[x] is the set of all polynomials in x with coefficients in R, with the usual rules for polynomial addition and multiplication: if f(x) is as above, and then g(x) = b m x m + b m 1 x m 1 + + b 1 x + b 0, f(x) + g(x) = max{m,n} k=0 k=0 i+j=k (a k + b k )x k, where we set a k = 0 if k > n, and b k = 0 if k > m, and m+n f(x) g(x) = a i b j x k, where we restrict i, j to be nonnegative and at most n, m respectively. 1
Definition. If f(x) = a n x n + + a 0 R[x] is not equal to 0, let i be maximal so that a i 0. Then the degree of f(x) is equal to i, and the leading coefficient of f is equal to a i. We say f(x) is constant if it is equal to 0, or has degree 0. Definition. A polynomial f(x) is monic if (it is not 0 and) its leading term is 1. Definition. A map ϕ : R R is a homomorphism if ϕ(1) = 1, and for all a, b R, we have ϕ(a + b) = ϕ(a) + ϕ(b) and ϕ(ab) = ϕ(a)ϕ(b). Definition. Given ϕ : R R a homomorphism, and α R, the homomorphism R[x] R mapping x to some α and mapping r R to ϕ(r) is called the evaluation homomorphism determined by α. If f(x) is a polynomial, we write f(α) for the image of f(x) under the homomorphism corresponding to α. Definition. Given a homomorphism ϕ : R R, the kernel is {α R : ϕ(α) = 0}. Definition. An ideal of a ring R is a nonempty subset I such that: (1) For all a, b I, we have a + b I. (2) For all a R and b I, we have ab I. Definition. If a R, we have the ideal consisting of the multiples of a. This is denoted (a), or ar. Ideals of this form are called principal ideals. Thus, the ideal which is all of R is called the unit ideal. The principal ideal (0) contains only 0, and is called the zero ideal. An ideal is called proper if it is not the unit ideal or the zero ideal. Definition. Let F be a field, and f, g F [x] polynomials, not both equal to 0. The greatest common divisor d F [x] of f and g is the unique monic polynomial which generates the ideal (f, g). It satisfies: (1) dr = fr + gr. (2) d divides both f and g. (3) If a polynomial h divides both f and g, then it divides d. (4) There are polynomials p and q such that d = pf + qg. Definition. A homomorphism is an isomorphism if it is bijective. Definition. If R is a ring, and R R a subring, and g 1,..., g n R, the ring generated by the g i over R, denoted R [g 1,..., g n ] is the smallest subring of R which contains R together with the g i. Definition. Let f(x) R[x] be a polynomial with the property that no nonzero constant is a multiple of f(x). To adjoin an element α to R, set where α denotes the congruence class of x. R[α] := R[x]/(f(x)), Definition. A ring R is an integral domain if for all nonzero b, c R, the product bc is also nonzero. Definition. Let R be an integral domain. A fraction is a symbol a/b or a b with a, b R, and b 0. Two fractions a/b and c/d are equivalent if ad = bc. We can add and multiply fractions as follows: a b + c ad + bc a =, d bd b c d = ac bd. Definition. If R is an integral domain, the field F constructed out of R using equivalence classes of fractions is called the fraction field (or field of fractions) of R. 2
Definition. An ideal I R is maximal if the only ideal of R strictly containing I is the unit ideal. Definition. An ideal I R is prime if, for all a, b R such that ab I, either a I or b I. Key results. Proposition. Let R be a ring, and f(x), g(x) R[x], with g(x) monic. Then there exist q(x), r(x) R[x] such that f(x) = g(x) q(x) + r(x), and either r(x) = 0 or the degree of r(x) is strictly less than the degree of g(x). Proposition. Let R, R be rings, and ϕ : R R a homomorphism. (1) For any α R, there is a unique homomorphism Φ : R[x] R that is equal to ϕ on the constant polynomials, and sends x to α. It is given by the formula (i 1,...,i n) a n x n + + a 1 x + a 0 a n α n + + a 1 α + a 0. (2) More generally, for any α 1,..., α n R, there is a unique homomorphism Φ : R[x 1,..., x n ] R that is equal to ϕ on the constant polynomials, and sends x i to α i for each i = 1,..., n. It is given by the formula a (i1,...,i n)x i 1 1 x in n a(i 1,..., i n )α i 1 1 αn in. (i 1,...,i n) Corollary. Given a polynomial f(x) R[x] and α R, then f(α) = 0 if and only if f(x) is a multiple of x α. Proposition. The ideals of Z are precisely nz for n 0. In particular, every ideal of Z is principal. Proposition. Let F be a field. Then every ideal I of the polynomial ring F [x] is principal. More specifically, if I (0), then I = (f(x)), where f(x) is an element of minimal degree in I. Theorem. Given an ideal I of a ring R, the quotient group R/I has a ring structure induced by multiplication in R. The canonical map π : R R/I is a ring homomorphism, and its kernel is I. Theorem (Correspondence theorem). Let ϕ : R R be a surjective ring homomorphism with kernel K. Then there is a bijective correspondence between ideals of R, and ideals of R containing K. This bijection is induced simply by taking images and preimages of ideals under ϕ. In addition, if I R is an ideal containing K, and I its image in R, then ϕ induces an isomorphism R/I R /I. Proposition. Let R be a ring, and f(x) R[x] a monic polynomial of degree n > 0. Let R[α] be obtained by adjoining a root α of f(x) as above. Then: (1) Every element of R[α] can be written uniquely in the form c 0 + c 1 α + + c n 1 α n 1 for c 0,..., c n 1 R. (2) Addition of two elements in R[α] corresponding to adding (c 0,..., c n 1 ) like vectors. (3) Multiplication in R[α] is calculated as follows: given β 1, β 2 R, if g 1 (x), g 2 (x) R[x] are the unique polynomials of degree less than n (if nonzero) such that g 1 (α) = β 1 and g 1 (α) = β 2, then divide g 1 (x)g 2 (x) by f(x): g 1 (x)g 2 (x) = f(x)q(x) + r(x) where r(x) = 0 or deg r(x) < n. We then have β 1 β 2 = r(α). 3
Theorem. Let R be an integral domain, and let F be the set of equivalence classes of fractions of elements in R. (1) Using addition and multiplication as defined above, F is a field. (2) R is a subring of F via the map induced by a a 1. (3) If R is a subring of a field K, then the rule a/b ab 1 defines an injective homomorphism of F into K. 2. Chapters 12 and 13: Definitions and results Sections covered. 12.2, 12.3, 12.5, and parts of 13.1, 13.4, and 13.5. Definitions. Definition. Let z be an element of an integral domain R. We say z is irreducible if it is not a unit, and for any factorization z = xy in R, either x or y must be a unit. We say z is prime if (z) is a prime ideal: equivalently, if z is not a unit, and if z divides xy, then z divides x or y. Definition. We will say a divides b in R if b is a multiple of a. Definition. A principal ideal domain (or PID) is an integral domain such that every ideal is principal. Definition. An integral domain R is a Euclidean domain if there exists a size function σ : R {0} Z 0 such that we can carry out division with remainder with respect to σ. That is, for any a, b R with b 0, there exist q, r R such that and either r = 0 or σ(r) < σ(b). a = bq + r Definition. Let R be a UFD, and f(x) R[x]. We say f(x) is primitive if there is no irreducible element p R which divides f(x). Definition. Let I, J be ideals of a ring R. Then the product ideal IJ is the ideal generated by elements of the form ab, with a I and b J. Explicitly, n IJ = { a i b i : a i I, b i J}. i=1 Definition. Let F be a subfield of a field K, and α K. We say α is algebraic over F if there exists a polynomial f(x) F [x] such that f(α) = 0. If α is algebraic, then the evaluation homomorphism F [x] K sending g(x) F [x] to g(α) has a nonzero kernel. But since F [x] is a PID, this kernel is principal, say (h(x)) for some monic polynomial h(x) F [x]. We call h(x) the minimal polynomial of α over F. Definition. α C is an algebraic integer if it is algebraic over Q, and the coefficients of its minimal polynomial are in Z. Definition. Let α C be algebraic. Then we also write Q(α) for Q[α]. The subset R of algebraic integers in Q(α) is called the ring of integers of Q(α). 4
Key results. Theorem. A principal ideal domain is a unique factorization domain. Corollary. Z is a unique factorization domain. If F is a field, then the polynomial ring F [x] is a unique factorization domain. Corollary. If R is an integral domain, and f(x) R[x] is nonzero, then f has at most deg f roots in R. Proposition. Z[i] is a Euclidean domain, with σ defined by σ(a + bi) = (a + bi)(a bi) = a 2 + b 2. Proposition. Every Euclidean domain is a principal ideal domain, and hence a unique factorization domain. Theorem. If R is a UFD, then R[x] is also a UFD. Corollary. Z[x 1,..., x n ] is a UFD, and F [x 1,..., x n ] is a UFD for any field F. Theorem. Let p be a prime number. Then there exist integers x, y with x 2 + y 2 = p if and only if either p = 2 or p 1 (mod 4). Proposition. Given a nonzero square-free integer d 1, let F = {a + b d : a, b Q}. Then F is a field. Let R be the subset of F consisting of algebraic integers. Then R is a ring, and it is equal to {a + bω : a, b Z} { d : d 1 (mod 4) where ω = 1+ d 2 : d 1 (mod 4). Theorem. Let α C be algebraic. Then the ring Q[α] is a subfield of C, which we denote by Q(α). The subset R of algebraic integers in Q(α) is a ring. called the ring of integers of Q(α). Theorem. Let R be the ring of integers of a field Q(α). Then R is a UFD if and only if it is a PID. Theorem. Let R be the ring of integers of a field Q(α). Then every ideal in R can be factored as a product of prime ideals, and this factorization is unique up to reordering. 5