Midterm2Solns - Math 310 Section 3.4 Fall 2010 Midterm 2...

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Unformatted text preview: Math 310 - Section 3.4 Fall, 2010 Midterm 2 r-o'blem 0mm Your Score! 1 10 ' .2: 15 g :3: 15 4 10 15' By Signing below I certify that I am striving to adhere to the. principles of acadeni'xic tmth and infiegriiy heid by Michigan State University: and S‘éit forth in the: Spartan Life Student Handfimok. Signature: Mafia 310 — Section 3. F‘rfllg 2010 B-fidterm 2 1. E10 ptsi; Comflete the definition 01 the term in-bold: A ring is a 110116:I1pt§-' Set 1? equipped with We operationsg 4:» and that Sgtisfy the. foiiowing axiums for all a. b: a E, R: Lwaflpéfl _ (in azi-"iaé—li; 2, m1 in-L“) "a {a “’3”; 1 MM m (awe: .: ‘ an: . 3.. as. 53%} b a“ I 3* fi(b+fi) x: g i3 33 6i C L“? § €13? ("JR 6a 4 63K * W W a" WM" mag , . K _ ‘ ‘ . W ML Mfg} (JUNE my 11W- a 9me MR ((g A)“; 2 ac ? be 2. ptb‘é Dufinc‘. :31 MW ackii‘sien and lxmitipficaflon on Q by ‘r‘ 2 9' + 5 + l and T 67> 5: 2 725 + ‘r + :3. $9170er that the Distributive Laws hold. for Q3 with those. operations L€.7% (xxég‘jic' 6Q ' “W b (ii—7:9} :2 dflfigih -% z: + E) €{55'i‘fi‘1’g} "3‘ “F” Elbmfl} m5 4&9 mg; : (aw- ca #2) 63" (m1 "F ‘3) Gib-M*Efiffiii'm‘m“i? flak (figfifiéfi "3- {dm—bflwflflfi "3(61‘fbfl3a, Mai-EM} '91.; . a; (gag: .59 E25}; :(_5§<; «é-aL-mjfigfim + hm") “r— {Lfi‘fflufllfi é~{$~m-i—b M1} ‘5» iii. {‘0} Does Q witi.1_ the. 0pcmtim1s above haw: a {faultiplicative identity? 131mm your answer is correct. U ‘fl‘h wnJHi¥i§7gfLsé§ifi ’EfiiéN'EW-‘fii‘f f ROME V‘ig 5% (53% C) '4‘ CH) + a +3 53L 0am a: {21%. "imam m «5a,. Brim}: 31.8 — Section 3. Fall, Z3010 Midterm 2 1:3. [15 pm] Which of the finlluwing are subrings of MEUJRJE’ Which ones have an identity? Prove. your ‘(lflfriW'Gl'S {UK-B COY'I'ECK \ ‘ ‘ ,. ‘O r (a) A}? matrices; of the Ionzn ( x U U > Wlth 7' t: a r" {635" m '5) wag" \ M 'H x “’éxszji) "” a 5 "5 m Misfit“ “Hm WEN”? Em") hf; ng'ii'iffimf'ffl, {a} an ((3 §> 1 (a {:7 \J E} R “gilt Euwh Wa fijyémfiflf , a _. _ _ D ‘ ‘ v ix \ \ ‘ ‘ { g) W?“ {a WEN}? Effimmfl g? “5’ M £3“ E} Mm? x a w '0‘". 535‘ 3: ‘9“): "pm M»; g 3M; a ax I ,}(§@) ((31.3? %{0€) ) With a, b E R. """"""""" “WEMWW ''''''' MTWWTH‘T‘M‘WW'MW > m. ML W 31W? 1”“ “9 WWW" MW Ii F) _ a: 41 L’ a; '61 «C ii "<1 _ R ‘.,__ r i u u ‘ , \ ( y fl ‘5 E) 3‘ §“ir?5fi” (Cl-1 52;} £49,”; 3} E: (:fl‘} m (:33; r‘“ (a w w? i} a») =3: (b) A1} Hiatricea of the form b5 it 32b {2%, H E}! 3’3 HM i“) R 5‘ Ergn‘ “ “‘ “‘ ~ A. «(it «w 6» E} iié‘i‘gwl i) 3)) “ion E Eric: anfigiwgfl / a {3 \ mmmmmmmm MWMMWM_W WWWWWWW _ mm with a. E : : r I “1 § a {31) T3133; swbfiwfi Ag) 5" ihvf'fi‘ékfl’kflhffvi re a. ‘ j? G" W Kai—i} : .9; a ‘ 1‘ ‘ . I {E ' {19% 'fi £224; a)» flflt’fin a? (5533‘ beiwga: paying, ((1) AR mini"in of the form \ {a 0. mm J “M (mwm 5‘: “fix (a :1 It}? {3:} 3T 11w.ku fix}; m3? {3 a gwfiwfflgu (a g.“ {3 5i 0 " 4. {M} pm} Lei. R be a. ring and a E R. 191543111119 that u ,i' 052, and film a. is not: a zero divisor. Prove fliat Whmilevc‘il‘ (L?) r (u; in R, then b m c. I fi'v‘fz‘wfi ‘37 333’ “1% Q“ :3 w? a gm fiffijéfl “ELM «cab Mags; 3: £133: Em Wfifig 2%) {dig-mg} “35%” {a Mag“. Math 3110 ~ Sectitm 23. Fail, 2010 i\-€Iidterm 2 5. pts.] (a) Let f : R —+ S‘ be a humoxxmrphism of rings. if v" is 21: zero (Ii'iViSOI' in R.‘ is f a zero divisor in S"? Prove you}? answer is mired. ML Ifiji'éfifilifii” j JLa‘rwé lax? 53‘) 3' {Jr [email protected] ilk-3f au- {14% a £32m dig/{W in bug“. 110 E5 mm“ m gerwfivfiw {A (3‘)) Let f : H M; S be an iSOlIM‘H‘phiSII‘: of ring& If 7" is; a zero divisor in R. is fir} a zero divisor in S? Provo. your answer is C(H‘T‘t‘l‘ib. . . - ,5 m K‘ i; a n: {Sim-Um" I‘m “‘5” {f‘jfifig‘ W 5&1” Siam. é} Efiéfidiw QM} fig) 2’ {35, #1 “ME mava a}; M £0”) HW) ":1 H?“ W "9 (0M “:2 Os; 3:3 41?} g; {9rd} d‘tv'éjw En- 5. [1'5 ptsj Prove-2 that. the rings 2724 and 23 >< Z23 are not isomorphic. I 5V??35L : 2%? W) g2 5 Es: WW" “"4" fimw?% Em ' "Tva 3‘? ‘Tkew m EE‘EflE a; [3% J CAE EJ’HWA. :3 j w Risa 1? 531%) £sz + Mg {figmg kg. Ham m «‘e Eva Engaw, t (_ mg, :43“) Ma W 595‘ '§}czw’t}>jnijm QKF}‘%~§, :1 L F [Dagny # Math .310 — Section 3., 53,11: ZUIU Midfierm f2. 7. [10 Prove the: fuliowing theorem: If R is an. il‘ltt-zgral (1031mm and fit} and 54,13) are. nonzero gmlynouflais in than d(’._(](f{i2f) : (i{,:g(f{;r;)) + [Zamgffly N- :2 \ 7‘ P I I. {:5}? WE. I; Ch + Gin GE; ’51 ‘ "’ “i” an ) CRY“ $ Uri“ 5M be -x~ bi {\4 'f b; 4:1 i- W -% 1% 5g ’5‘ I gym Twig; " NH“: 71%, "1 qt» b‘fi + (GR-1b; 4” qt billy: ‘5‘ “" 'i“ ("in 5m) it 313% R S: 6m r3}?er 610M“, 0% bmyrog‘, is f}'}‘I’lr~i J’ lacuna CG-a'é:Rt/fm+ O‘g '€(“}fi’d} [\J 5% hm 2 KL Cbfi'm’f'fi‘cfén+ Vii, '1 W 8. [11} 33%.] L01: f5}:‘).h,(:r:} wlilerE} F is a field, anti flat) and y(:!:) are relatix’eziy' prime. If hfn’tflf prmre that M37) and am I‘elaxiveiy prime. 5m WWM mm»: W, “m w Wm “my ‘Ac-Lmem 7 CM um + 3M V65} F M '3 4M 6 'FQ] 53., Em a W) m)” SUM-«J-i-W’rhj Hm M.) 1%. Wm ei‘fwffmj m m m) 1+ W m) if: TM MM (WW5) 4- «awmvwfl his ‘1“‘meg We; +23- %) 5M m mle WM". ...
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