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12. Let a and b be elements of a ring R. (a) Prove that the equation a + x = b has a unique solution in R. (You must prove that there is a solution and that this solution is the only one.) (b) If R is a ring with identity and a is a unit, prove that the equation ax = has a unique solution in R. b

12. Let a and b be elements of a ring R. (a) Prove that the equation a + x = b has a unique solution in R. (You must prove that there is a solution and that this solution is the only one.) (b) If R is a ring with identity and a is a unit, prove that the equation ax = has a unique solution in R. b

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Thomas W. Hungerford - Abstrac x + O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201.. ... Flash Player will no longer be supported after December 2020. Turn off Learn more of 621 + -- A' Read aloud V Draw F Highlight O Erase 89 numbers is a subring of M(R). 10. Let R and S be rings and consider these subsets of R X S: R= {(r, Os) |rER} and S= {(OR, s) | SES}. %3D (a) If R= Z3 and S = Zs. What are the sets R and S? (b) For any rings Rand S, show that Ris a subring of RX S. Goun at (c) For any rings R and S, show that S is a subring of RX S. 11. Let R be a ring and ma fixed integer. Let S = {rɛR|mr = is a subring of R. OR}. Prove that S 12. Let a and b be elements of a ring R. (a) Prove that the equation a + x = b has a unique solution in R. (You must prove that there is a solution and that this solution is the only one.) (b) If R is a ring with identity and a is a unit, prove that the equation ax = b has a unique solution in R. 13. Let Sand T be subrings of a ring R. In (a) and (b), if the answer is "yes," prove it. If the answer is “no," give a counterexample. (a) Is Sn Ta subring of R? (b) Is SU Ta subring of R? Copyrigt 2012Ca ng AN Righ Ra nd May aot be apied ed ar dapticd is whele ar ta part D to dearonie cig, tird pary eate mey be ned fon te eBoak darehagerta). Bd N ba ned ttoy apped a d dly dea be ovd ing apria Cagge Loaing a eright to ove dddonnl a ty timef D dghts cetti nt 9:06 AM O Search for anything EPIC Ai EPIC O 4) 34 12/7/2020
Transcribed Image Text:Thomas W. Hungerford - Abstrac x + O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(201.. ... Flash Player will no longer be supported after December 2020. Turn off Learn more of 621 + -- A' Read aloud V Draw F Highlight O Erase 89 numbers is a subring of M(R). 10. Let R and S be rings and consider these subsets of R X S: R= {(r, Os) |rER} and S= {(OR, s) | SES}. %3D (a) If R= Z3 and S = Zs. What are the sets R and S? (b) For any rings Rand S, show that Ris a subring of RX S. Goun at (c) For any rings R and S, show that S is a subring of RX S. 11. Let R be a ring and ma fixed integer. Let S = {rɛR|mr = is a subring of R. OR}. Prove that S 12. Let a and b be elements of a ring R. (a) Prove that the equation a + x = b has a unique solution in R. (You must prove that there is a solution and that this solution is the only one.) (b) If R is a ring with identity and a is a unit, prove that the equation ax = b has a unique solution in R. 13. Let Sand T be subrings of a ring R. In (a) and (b), if the answer is "yes," prove it. If the answer is “no," give a counterexample. (a) Is Sn Ta subring of R? (b) Is SU Ta subring of R? Copyrigt 2012Ca ng AN Righ Ra nd May aot be apied ed ar dapticd is whele ar ta part D to dearonie cig, tird pary eate mey be ned fon te eBoak darehagerta). Bd N ba ned ttoy apped a d dly dea be ovd ing apria Cagge Loaing a eright to ove dddonnl a ty timef D dghts cetti nt 9:06 AM O Search for anything EPIC Ai EPIC O 4) 34 12/7/2020
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