b2ring2 - B2 ring theory preliminaries Michael Vaughan-Lee...

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Unformatted text preview: B2 ring theory preliminaries Michael Vaughan-Lee September 2000 1 Introduction The standard way of writing an algebra textbook is to first give the formal defini- tions of groups or rings or fields or whatever, and then look at some examples later. The difficulty with this approach is that the formal definition does not make any sense to anyone who doesnt already know what a group or a ring or a field is. So I am going to start with some familiar examples of rings. The ring of integers Z , The ring Z [ x ] of polynomials with integer coefficients, The ring K [ x ] of polynomials with coefficients from a field K . The ring K [ x,y ] of polynomials in two variables, The ring M n ( R ) of n n matrices with entries in a ring R , The Gaussian integers Z [ i ] = { m + ni | m,n Z } (where i 2 =- 1). The key property of these algebraic structures is that you can add and mul- tiply elements. But in general you cannot do division: if p ( x ) and q ( x ) are two polynomials then in general p ( x ) /q ( x ) is not a polynomial. Now for the formal definition. Definition 1 A ring is a set R with two binary operations + , (plus and times), a unary operation- (minus), and a zero element . (We usually write ab for a b . Sometimes we write a.b .) These operations must satisfy the following axioms: 1. ( a + b ) + c = a + ( b + c ) for all a,b,c R , 2. a + b = b + a for all a,b R , 3. a + 0 = 0 + a = a for all a R , 4. a + (- a ) = (- a ) + a = 0 for all a R , 1 5. ( ab ) c = a ( bc ) for all a,b,c R , 6. a ( b + c ) = ab + ac for all a,b,c R , 7. ( a + b ) c = ac + bc for all a,b,c R . The first four axioms imply that R is an abelian group under the operations of + ,- , 0. Axiom (5) says that multiplication is associative, and axioms (6) and (7) are the two distributive laws. The ring axioms do not require that multiplication be commutative. For example multiplication of matrices is not commutative. However we will be working with commutative rings with unity. Definition 2 The ring R is commutative if ab = ba for all a,b R . Definition 3 We say that R is a ring with unity if there is an element 1 R such that a. 1 = 1 .a = a for all a R . Definition 4 R is an integral domain if R is a commutative ring with unity and if the product of two non-zero elements of R is always non-zero (in other words ab = 0 implies a = 0 or b = 0 ). 2 Subrings, homomorphisms and quotient rings A subring is a subset which is closed under the ring operations, and a ring homo- morphism is a map which preserves the ring operations. You can work out what the detailed definitions are without ever having seen them before! But here they are anyway....
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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b2ring2 - B2 ring theory preliminaries Michael Vaughan-Lee...

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