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Linear Algebra

by: Maximilian Turcotte

Linear Algebra MATH 214

Maximilian Turcotte

GPA 3.7


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This 5 page Class Notes was uploaded by Maximilian Turcotte on Monday October 5, 2015. The Class Notes belongs to MATH 214 at Colgate University taught by Staff in Fall. Since its upload, it has received 30 views. For similar materials see /class/219588/math-214-colgate-university in Mathematics (M) at Colgate University.

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Date Created: 10/05/15
Section 18 Ring Homomorphisrns Let s make it of cial Def A function 4p from one ring R to another S is a ring homomorphism iff it respects the ring operations For all 011 6 R Ma b Ma W and ab WWO Because a ring homomorphism is rst an additive group homomorphism we have MOB 05 and for n in Z and a in R 4pna mpa And it is also true that Ma Ma for any positive integer 71 But weird things can happen with unities if they exist at all Note rst that if R is a ring with unity and 4p R H S is a ring homomorphism for which 4p1R 05 then for all r in R ltpr 4pr1R 4pr4p1R 4pr05 05 so 4p is the constant function 0 0 Ex Consider the function 4p IR H M2X2IR r H lt 6 0 gt This is a ring homomorphism and both rings have unities 1 and gt respectively but the homomorphism doesn t take the 1 0 0 1 unity of IR to the unity of M2X2IR Exs o For any positive integer n the function Z H Zn x H z mod n is not just a homomorphism of additive groups it is also a ring homomorphism 0 Complex conjugation sending each complex number 2 z I yi where zy 6 IR to its complex conjugate E x H yi turns out to be a ring automorphism of C Similarly in the ring a I bx2 a b E Z the function a I bx2 H a H bx2 is an automorphism of o If R S are rings and R x S is the direct product of rings with coordinatewise operations the projection onto the rst coordinate R x S H R r s H r is a ring epimorphism H as is the projecton onto the second coordinate S The inclusion as the rst coordinate R H R x S r H r 0 is a ring monomorphism as the inclusion as the second coordinate If R S have unities then R x S has unity 13 15 and the projections take unity to unity but the inclusions don t do that o If a is a xed element of the set X and fX is the family of all functions X H IRwith pointwise operations then the evaluation function 6a fX H IR f H fa is a ring epimorphism Similarly on the set IRIm of polynomials in the variable x with real coef cients if a is a xed real number the evaluation function which I will again denote 8a IRle H IR pz H pa is a ring homomorphism H the de nitions of addition and multiplication of polynomials which look weird in the abstract ask a struggling high school algebra student were chosen to make that work 0 On the set IRIm of polynomials in the variable x with real coef cients differentiation is a homomorphism of additive groups but it is Lot a ring homomorphism because the product rule is not Dug DfD9 o On the set M2X2IR of 2 x 2 matrices with real entries the determinant function onto IR respects multiplication but not addition so it is not a ring homomorphism And the trace function M2X2IR H IR which we don t usually mention in our Math 214 H it s just the sum of the main diagonal entries respects addition but not multiplication so it is not a ring homomorphism either 0 Consider the functions 2 H Zn given by x H am where a is a xed element of Zn and the last multiplication is mod 71 They are all additive group homomorphisms they are epimorphisms and hence automorphisms exactly when a is a generator of Zn ie relatively prime to 71 Usually though they are not ring homomorphisms because they don t respect the multiplication They only do that ie they are only ring homomorphisms if a is an idempotent a2 a We always have the idempotents 0 and 1 of course giving the zero function and the identity function on Zn others are 3 or 4 in 26 4 or 9 in 212 etc To get an idempotent other than 0 or 1 in Zn we need 71 to divide aa 7 1 for some a between 2 and n 7 1 Theorem 182 says that if R S are rings with unity and 4p R a S is a ring homomorphism for which 4p1R 7 05 then 4p1R 15 provided S is either a division ring or an integral domain The two cases fall into a single one 4p1R 15 provided S has no nonzero zero divisors Part of the same theorem 182 can be stated a bit more strongly If R has a unity and 4p R a S is an epimorphism then 4p1R is a unity for S Let s collect the basic facts about ring homomorphisms Prop Let 4p R a S be a ring homomorphism o If A is a subring of R then 4pA is a subring of S If A is an ideal in R 4p is onto S then 4pA is an ideal in S If B is a subring of S then 4p 1B is a subring of R If B is an ideal in S then 4p 1B is an ideal in R o lf 1 S gt T is another ring homomorphism then 1 0 4p R a T is a ring homomorphism lf 4p is a ring isomorphism then so is Ap l The kernel A r E R ltpr 05 of 4p is an ideal in R and the canonical group homomorphism R a RA is a ring epimorphism Fundamental Theorem of Ring Homomorphisms Again let A kerltp The group isomor phism E RA a MR Ar H ltpr is also a ring isomorphism All of this is easy to check Let s just give a quick example to show why we need onto for the image of an ideal to be an ideal The set 22 of even integers is an ideal in Z and the inclusion function Z a Q z gt gt z is a ring homomorphism but in Q 22 no longer captures multiplication 1 7 2 1 22 2 Cor Let R S be rings with unity and 4p R a S be a ring homomorphism for which 4p1R 7 05 Then a For every unit u in R Mu 7 05 In particular any ring homomorphism from a division ring or eld to some other ring either takes every element to 05 or is a monomorphism b If r s in R satisfy rs OR and ltpr is a unit in S then 4278 05 Pf a MWMU I M112 7g 037 SO 0075 OS b 8 wr 1wrws WWW WTlOs 03 Prop and Def Let R be a ring with unity a The function 4p l a R n gt gt n1R is a ring homomorphism the image Ml is in the center of R the set of elements that commute with every element of R and is called the prime subring of R the nonnegative generator of the kernel of 4p is the characteristic of R b If R has no nonzero zerodivisors then the additive order of 13 which is the characteristic if the characteristic is nonzero and in nite if the characteristic is 0 is also the additive order of every nonzero element of R and ifit is nite then it is a prime number p so that Ml E lp a eld A O V Suppose R is a division ring if it has nite characteristic we have just seen that it contains a copy of some lp If it has characteristic 0 then 4p is a monomorphism of l into R and mlRn1R 1 77171 6 l n 7 0 E Q is a sub eld in the center of R The sub eld congruent lp or Q depending on the characteristic is called the prime sub eld of R Pf a The elements 71113 as integer multiples of 1R must commute with every element of B so they are in ZR It is easy to check that 4p is a ring homomorphism The rest of part a is de nitions b Let r be a nonzero element of R and suppose nr OR for some positive integer Then rnlg OR and because R has no nonzero zerodivisors n13 03 Conversely if n13 03 then it is easy to see that nr OR for all r in B So the additive orders of all the nonzero elements of R are equal Now suppose that order is nite but not prime say it is mn where m n are integers greater than 1 Then 0 mn1R m1Rn1R and because R has no nonzero zerodivisors one of m1Rn1R must be 0 but that means the additive order of 13 is less than mm the desired contradiction So Ml is isomorphic to lp for some prime p c In characteristic 0 Because Ml Q ZR the reciprocals of its nonzero elements are also in ZR So mlRn1R 1 m n E l n 7 0 is in ZR and is a eld isomorphic in the obvious way to Q The rest of part c is de nitions The text proves in some detail the following fact Prop and Def Let R be an integral domain Then there is a eld F called the eld of fractions of R which contains an isomorphic copy of R and such that every element of F can be written in the form alf1 where a b E R and b 7 0 It is only a bit messier and actually clearer to prove a more general result Prop and Def Let R be a commutative ring and S be a subset of R that is closed under multiplication for simplicity assume R has unity and 13 E S Then there is a ring S lR with unity called the ring of fractions ofR with respect to S such that i there is a ring homomorphism 4p R a S lR that takes 13 to the unity of S lR ii for all s in S Ms is a unit in S lR and iii every element of S lR has the form MrMs 1 The kernel of 4p consists of the elements r of R for which sr 0 for some 3 in S Pf On the set R x S de ne the relation R by a sRb t iff there is an element a of S for which atu bsu If S contains no zerodivisors then we don t have to bother with the extra u and the 4p we get will be a monomorphism R oslla es in so os72o 5 lol all e 5 ln R x s s ll o 5Rbt then Mu bed so bsu Mu so home 5 T ll o s73bt and btRc n then am but and 7qu am lol some mm ln s so awtwu motm bsu39uw b39uwsu ctwsu cstmu and new 6 S because 5 ls closed undel multlplloatlon so o s72e 1 So 72 ls an equwalence lelatlon on R x 5 denote the equlyalenoe class ol os by os and the set ol all such equlyalenoe classes by Squot Thlnk ol os as os and the lollowlng de nmons wlll make sense e ne the opelatlons ol addltlon and multlplloatlon on these equlvalenoe classes os ht otbsst and doubt obst o oulse we need to check that these opelatlons ale wellrde ned ll d5 ow and ht bat1 say esn 2511 and we m then downt bt oss u Melon1 Vwss u W Vsst39uu at bss t ml and abswu emoteo e tnxl so st no whele an e s so etlsst a t Vs s t and elst MKS Thelelole the opelatlons ale wellrde ne We can check that thls addltlon ls oommutatlve assoolatlve and dlstllbutlve ovel the multiply oatlon that multlplloatlon ls assoolatlve and oommutatlve that 0B 1B ls azelo and 725 ls a negathe lol os and that 1 11a ls a unlty and 15 ls a multlpllcatwe mvezse lol 51la Final we need to de ne the homomolphlsm lp R a S R Set mo t 1R Then we can check that lp ls a llng homomolphlsm We have seen that lol sub 5 ln 5 lps 5 lla ls a unlt ln 5 R n element t olR ls ln the kezneloftplff71kUk1klelf flhexe ls an elements ol 5 lolwhloh 71R5 0R1Rs le l3 to 0R lol some 5 ln 5 The text makes the polnt that the eld ol llaotlons ol an lnteglal domaln R ls the umque smallest eld that oontalns R In the same way ll R ls a oommutatlve llng and 5 closed undel multlplloatlon and ll ls a oommutatlve llng wlth unlty lol whlch thele ls a llng p R s T wlth the plopelty that we ls a unlt ln T lol every 5 ln s then thele ls a llng homomolphlsm a Squot s T de ned by We 5 4494 lol whlch the Joy d l e the lollowlng dlaglam ls oommutatlve 4X R h T Squot may seem vely altl olal but lt ls used a gleat deal ln oommutatlve algebla Fol example ll P ls a pll e ldeal ln a oommutatlve llng R wlth unlty then R 7 P ls closed undel multlplloatlon so we can lolm the llng R 7 PquotR whlch ls usualh denoted RP and called the locallzallon olR at P l The set PRP 25 RP and every element outslde lt ls a unlt In the case R 1 an F pl whele p ls a pllm o m u m an l 5 E7 3 2 a g a 3 5 integer this ring would be the set of rational numbers where the denominator is not divisible by p In the case B Rmy and P is the set of polynomials that take the value 0 at a point ab in the plane RP is the set of rational functions quotients of two polynomials that are de ned in a neighborhood of ab the neighborhood varies with the function Etc On the other hand if we have an element 3 of a commutative ring R with unity and we want to see what happens if we give 3 a reciprocal then of course we are also making units of all the powers of s so we form the ring 3 n E 2 1R


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