Transcription of De nition and Examples of Rings
1 LECTURE 14 Definition and Examples of a nonempty setRequipped with two operations and (more typicallydenoted as addition and multiplication) that satisfy the following conditions. For alla, b, c R:(1)Ifa Randb R, thena b R.(2)a (b c) = (a b) c(3)a b=b a(4)There is an element0 RinRsuch thata 0R=a , a R .(5)For eacha R, the equationa x= 0 Rhas a solution inR.(6)Ifa R, andb R, thenab R.(7)a (b c) = (a b) c.(8)a (b c) = (a b) (b c) ringis a ringRsuch that( )a b=b a , a, b R . with identityis a ringRthat contains an element1 Rsuch that( )a 1R= 1R a=a , a R .Let us continue with our discussion of Examples of ,Q,R, andCare all commutative Rings with an interval on the real line and letRdenote the set of continuous functionsf:I be given the structure of a commutative ring with identity by setting[f g](x)=f(x) +g(x)[f g](x)=f(x)g(x)0R function with constant value 01R function with constant value 1and then verifying that properties (1)-(10) the set of continuous functionsf:R Rsuch that 0f(x)dx <.
2 4914. DEFINITION AND Examples OF RINGS50We can definef g,f g, 0 Rjust as in the previous example; however, we cannot define a multiplicativeidentity element in this case. This is because 01dx= limx (x 0) = so the function 1 Rof the previous example does not belong to this set. Thus, the set of continuous functionsthat are integrable on [0, ) form a commutative ring (without identity).Example the set of even a commutative ring , however, it lacks a multiplicativeidentity setOof odd integers is not a ring because it is not closed under the preceding example shows, a subset of a ring need not be a a subset of the set of elements of a ringR. If under the notions of additionsand multiplication inherited from the ringR,Sis a ring ( conditions 1-8 in the definition ofa ring ), then we saySis a subset of a ringR.]
3 ThenSis a subring if(i)Sis closed under addition.(ii)Sis closed under multiplication.(iii)Ifs S, then s R, the additive inverse ofsas an element ofR, is also axioms 2, 3, 7, 8 hold for all elements of the original ringRthey will also hold for any subsetS , to verify that a given subsetSis a subring of a ringR, one must show that(1)Sis closed under addition This is implied by condition (i) onS(4)Sis closed under multiplication; This is implied by (ii) onS.(5) 0R Sand (6) Whena S, the equationa+x= 0 Rhas a solution inS. If (iii) is true, then the additive inverse s Ralso belongs toSifs S. But thens+ ( s) =0R S, because by (i)Sis closed under addition. But thenOR+s=sfor everys S, and soORis the additive identity forS( ). So if (i) and (iii) are true, thenShas an additiveidentity and forSthen for everys Swe have a solution ofs+x= (Z),M2(Q),M2(R) andM2(C) denote the sets of 2 2 matrices with entries, respec-tively, in the integersZ, the rational numbersQ, the real numbersR, and the complex numbersC.
4 Additionand multiplication can be defined by(a bc d) (e fg h)=(a+b b+fc+g d+h)(a bc d) (e fg h)=(ae+bg af+bhce+dg cf+dh)witha, b, c, d, e, f, g, hin, respectivelyZ,Q,R, andC. The matrices0R=(0 00 0)1R=(1 00 1)14. DEFINITION AND Examples OF RINGS51are then, respectively, additive identity elements and multiplicative identity elements ofR. Note howeverthat(1 10 1)(1 01 0)=(2 01 0)6=(1 11 1)=(1 01 0)(1 10 1)so multiplication inRis not commutative in general. Thus, each of these sets is a non-commutative ringwith have seen that some Rings likeZorZpwithpprime have the property thata b= 0R a= 0 Rorb= 0R;but that this is not a property we can expect in general. This property is important enough to merit aspecial domainis a commutative ringRwith identity1R6= 0 Rsuch that( )a b= 0R a= 0 Rorb= that the ringZpwhenpis prime has the property that ifa6= [0], then the equationax= [1]always has a solution inZp.
5 This not true for the ringZ; because for example, the solution of2x= 1is12/ Z. However, the ringQof rational numbers does have this ringis a ringRwith identity1R6= 0 Rsuch that for eacha6= 0 RinRtheequationsa x= 1 Randx a= 1 Rhave solutions that we do not require a division ring to be a division ring with commutative the most part we will be concentrating on fields rather than non-commutative division :Q,R, :In the ringM2(C), let1 =(1 00 1),i=(i00 i),j=(01 1 0),k=(0ii0).The setHofreal quaterionsconsists of all matrices of the forma1 +bi+cj+dk=(a+ibc+di c+di a bi)wherea, b, c, d R. It is easy to verify thatHis closed under the usual addition of matrices. Also 1ijk11ijkii-1k-jjj-k -1ikkj-i-114. DEFINITION AND Examples OF RINGS52 Note that multiplication is not commutative in this ring ; ,ij=k= is possible to shownevertheless thatHis not only a ring with identity but a division that the Cartesian productA Bof two setsAandBis the set of all ordered pairs (a, b) witha Aandb Rings .
6 Define addition and multiplication onR Sby(r, s) + (r, s) = (r+r, s+s),(r, s)(r, s) = (rr, ss).ThenR Sis a a ring . IfRandSare both commutative, then so isR S. IfRandSeach has anidentity, then so doesR (homework problem)
