1 Commutative Rings with Identity

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Harold Bond
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1 1 Commutative Rings with Identity The first-year courses in (Abstract) Algebra concentrated on Groups: algebraic structures where there is basically one algebraic operation multiplication with the associated inverse and identity. Although apparently relatively uncomplicated they provided an example of the basics of algebra we discussed subgroups, quotient groups, homomorphisms of groups as well as discussing specifically group-y results like Lagrange s Theorem. In this course we want to look at rather richer structures, namely rings : the sort of structure where one can both add and multiply. In fact we will deal exclusively with an important subclass: we will only deal with commutative rings, where the multiplication obeys the law a b = b a; and we will only deal with rings for which there is a multiplicative identity, and element 1 satisfying the law 1 x = x 1 = x. Whenever you are in doubt about what a theorem means, then the example to keep returning to is the ring of integers, Z. 1.1 The Definition Definition 1. A commutative ring with identity is a non-empty set A, equipped with certain operations (see below) satisfying certain laws or axioms (see below) The operations The operations are as follows. For every pair of elements a, b A there is a unique element a + b, called their sum. For every pair of elements a, b A there is a unique element a b, called their product. For element a A there is a unique element a, called its negative. There is a special element 0 A called the zero. There is a special element 1 A called the identity element The axioms The following laws must hold for all elements a, b, c A: A1 a + b = b + a A2 a + (b + c) = (a + b) + c A3 a + 0 = a A4 a + ( a) = 0 M1 a b = b a M2 a (b c) = (a b) c M3 a 1 = a [+ is commutative] [+ is associative] [zero and addition] [negatives and addition] [ is commutative] [ is associative] [identity and multiplication] D a (b + c) = a b + a c [ distributes over +] Notation { ab for a b We write a b for a + ( b). 1

2 1.1.4 Comments Informally, in a commutative ring with identity we can add, subtract, multiply, and we have an identity. Our laws are satisfied in Z and capture (we hope) the algebraic essence of the integers. (What we have not attempted to build in is the order enjoyed by the integers.) Note that the Axioms (A1) (A4) tell us that our commutative ring with identity A is under addition an abelian group. We don t propose to repeat work done in the first year Groups course; for example we will use without fuss such facts as the zero element is unique. In a similar vein we will not repeat work done in the first year Analysis I course; much of what we did when we investigated the real numbers from an axiomatic point of view can be used here too. For example, we don t mean to fuss at all when we use facts like (b + c) a = b a + c a. We will also follow the practice we have learned in Groups and Vector Spaces: all zero elements will be denoted by 0, and all identity elements by An example We will deal with examples later, but here is an example rather different in flavour from the integers Z. For A take the set of diagonal n n matrices with real entries; for the operations take the usual matrix operations. Then we have a commutative ring with identity. 1.2 Two important classes We begin with two definitions. Definition 2. A non-zero element z of a commutative ring with identity A is called a zerodivisor if there exists a non-zero element w A such that zw = 0. For example, in the commutative[ ring with ] identity consisting [ ] of the 2 2 diagonal matrices with real entries the element and its friend are both zero-divisors [ ] a 0 More generally, the zero-divisors are precisely the with a = 0 or d = 0. 0 d Definition 3. An element u of a commutative ring with identity A is called a unit if there exists an element v A such that uv = 1. In this case we say that v is the inverse of u. We denote the set of units by A, which we call the (multiplicative) group of units of A. Note that if u is a unit then u is not a zero divisor. For a trivial example, in any A the identity is a unit. For a more elaborate example take again [ for ] A the 2 2 diagonal matrices with real entries. Then the units are precisely the a 0 with ad 0. 0 d Definition 4. Let A be a commutative ring with identity, and let a A. The elements a of a commutative ring with identity A are called an associate of a if for some unit u A we have that a = ua. Note that this is an equivalence relation. 2

3 1.2.1 Integral Domains We can now define this important class of rings. Definition 5. We say that a commutative ring with identity A is an integral domain if 1 0 and there are no zero-divisors. For example, Z is an integral domain; other examples appear below Fields Even more specialised are the fields. Definition 6. We say that the commutative ring with an identity is a field if 1 0 and every non-zero element is a unit. For example, R is a field. Note that this definition of field (setting fields in a more general picture) is completely consistent with the definition used in the Linear Algebra course. 1.3 Examples, Non-examples and Nearly Examples The integers We repeat: the integers, with the usual operations form a commutative ring with identity Some fields Here are examples of fields that we have met already: the rational field, Q, the real field R, the complex field C. There are other, more exotic fields: many we will construct later as examples of theorems we prove. For the moment you may like to check that Q[ 2] := {a + b 2 a, b Q} is a field; and that C(X) := { f(x) g(x) f, g polynomials with complex coefficients, g 0} is a field Polynomials over a field Let K be any field. Then the set of polynomials K[X], with the usual polynomial definitions of addition and multiplication forms a commutative ring with identity. Next to Z these are our most important examples of commutative rings with identity Not quite examples The set of n n matrices with entries from a field K is not a commutative ring with identity; it fails the commutative requirement. But suitably adapted much of what we say and prove can be adapted to this situation. Various subsets, however, yield useful examples. The set of even integers is not a commutative ring with identity; it fails the identity requirement. Again some of what we say can be suitably adapted to this sort of situation. 1 Z is what number theorists study, the polynomial rings are what the geometers study; the similarity between the structures goes very deep. 3

4 1.4 Subrings Definition 7. Let A be a commutative ring with identity. A subset B A is said to be a subring [more properly, a sub-(commutative ring with identity)] if is a commutative ring with identity under the same operations. For example, Z is a subring of Q. Just as for subgroups we have a Proposition (Test for Subringhood). Let A be a commutative ring with identity. Then B A is a sub-(commutative ring with identity) if and only if (i) 0 B; (ii) if a, b B then (a b) B; (iii) 1 B; (iv) if a, b B then ab B. Proof. The proof is just as for groups or vector spaces; these criteria guarantee that the operations restrict to operations on B and then the fact that the axioms which hold for all elements of A certainly hold in B. As an application: the only sub-(commutative ring with identity) of Z is Z. Note 1. Note that if by ring we mean, as most authors do, a system satisfying our axioms (A1) (A4) and (M2) and (D), then there are many subrings of Z: for each d Z the set dz := {dr r Z} is a sub-ring, but has no identity. It is therefore sometimes important to adopt the tedious sub-(commutative ring with identity) language. To continue the application: why is the commutative ring with identity {0}, which is a subset and a commutative ring with identity, not a sub-(commutative ring with identity) of Z? 1.5 Direct Products This is an easy recipe to make new rings from old. Proposition Let A 1 and A 2 be commutative rings with identity. Then the set A 1 A 2 := {(x 1, x 2 ) x i A i, i = 1, 2} is a commutative ring with identity under the coordinatewise operations: (i) the zero element is (0, 0); (ii) (a 1, a 2 ) := ( a 1, a 2 ); (iii) (a 1, a 2 ) + (b 1, b 2 ) := (a 1 + b 1, a 2 + b 2 ); (iv) the identity element is (1, 1); (v) (a 1, a 2 ) (b 1, b 2 ) := (a 1 b 1, a 2, b 2 ). Proof. Trivial. We usually denote this ring by A 1 A 2 (or sometimes just A 1 A 2 ). 4

5 For an example, take R R. This is a commutative ring with identity. Considered just as an additive group it is isomorphic to C; but as rings they are very different, C has no zero-divisors, but every 2 A 1 A 2 has lots. 1.6 Polynomial Rings This is another recipe to make new rings from old. We give it in quite a general form, in order to demystify the unknown. Proposition Let A be a commutative ring with identity. Then the set of sequences { } (a k ) k=0 : a k A, only a finite number of the entries a k non-zero is a commutative ring with identity under the operations: (i) the zero element is (0, 0,...); (ii) (a k ) k=0 := ( a k) k=0 ; (iii) (a k ) k=0 + (b k) k=0 := (a k + b k ) k=0 ; (iv) the identity element is (1, 0, 0,...); (v) (a k ) k=0 (b k) k=0 := ( r+s=k a rb s ) k=0. Proof. It is easy to check that the set is closed under the operations. The addition axioms are trivial. For this convolution multiplication the axioms are slightly tedious to check, but not difficult. What has this got to do with polynomials in X? Well, write X := (0, 1, 0,...0), and note that X 2 = (0, 0, 1, 0,...) and so on. With that in place we then recover all polynomials: for example, (a 0, a 1, a 2, a 3, 0, 0,...) = a 0 + a 1 X + a 2 X 2 + a 3 X 3. With this choice of name X for (0, 1, 0, 0,...) we call this new ring A[X]. If we called (0, 1, 0,...) by the name Y we d call the new ring A[Y ]. We can repeat the process, and manufacture for example R[X][Y ], which we usually abbreviate to R[X, Y ]. The study of real plane curves is essentially the study of this ring Power Series Rings We can perform the same construction on the set of all sequences. In that case we get the power series ring denoted by A[[X]]. Note that this is algebra, there s no question of convergence. 1.7 Important: Notation All the rings in this course are commutative rings with identity. We will from now on usually just say ring. We will say subring and mean sub-(commutative ring with identity) and later we will speak of ring homomorphism and mean homomorphism of commutative rings with identity And so on. 2 Well, almost every... 5

ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.

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