5

PROBLEM 1Let R be a ring and let Hom(R,R) denote the set ofall additive maps from R to R Hom(R,R) has two operations defined on it:That is: f € Hom(R,R) if f(...

Question

PROBLEM 1Let R be a ring and let Hom(R,R) denote the set ofall additive maps from R to R Hom(R,R) has two operations defined on it:That is: f € Hom(R,R) if f(a + b) = f(a) + f(b). So to check a map is in Hom(R,R), you only have to check it preserves addition:For f,g € Hom(R, R) defineAddition by (f + g)(r) = f(r) + g(r) forallr € R andMultiplication is composition, that is (f 0 g)(r) = f(g(r)) for allr € RIt turns out that Hom(R,R) is a ring " under these two operations The point of

PROBLEM 1 Let R be a ring and let Hom(R,R) denote the set ofall additive maps from R to R Hom(R,R) has two operations defined on it: That is: f € Hom(R,R) if f(a + b) = f(a) + f(b). So to check a map is in Hom(R,R), you only have to check it preserves addition: For f,g € Hom(R, R) define Addition by (f + g)(r) = f(r) + g(r) forallr € R and Multiplication is composition, that is (f 0 g)(r) = f(g(r)) for allr € R It turns out that Hom(R,R) is a ring " under these two operations The point of this problem is to check a few of the ring properties: Show Hom(R,R) is closed under addition (as defined above): that is, for f,9 € Hom(R,R), prove f + g € Hom(R,R),i.e the sum of two ring homomorphisms is ring homomorphism Show Hom(R,R) is closed under composition (as defined aboveJ: that is, for f,9 € Hom(R,R) prove f 9 € Hom(R,R),ie_ the composite of two ring homomorphisms is a ring homomorphism_ Define id: R v Rby id(r) for allr € R. Prove that id is the multiplicative identity element; that is f id = id f =f forall f € Hom(R,R) Prove the following distributive law: f (g + h) = f 0 g + f h, for all f,g,h € Hom(R,R) (apply both sides tor € R and check you get the same answer):



Answers

Let f be an ordered field and x,y,z in F.
Prove that if x<0 and y<z, then xy>xz.

So to show that this is not a hormone dwarfism. Um We essentially just have to show that one of these statements here is not true because these are the two criteria for something to be a home abort is um for some function and for this, I think the bottom one will be the easiest to actually go about doing. So How I'll do this is let's first plug in. Just half of zero or a half of the zero factor, I should say. So 00. And if we were to plug that in, so this is going to be zero, this is going to be zero and then that's going to be negative too. So we get that this outputs negative too. Well, if this is a home of dwarfism, if I were to multiply the zero vector by anything, then it should give me the same output as if I were to just multiplied on the inside and then go about doing So Let's say I were to just multiply the outside of this by two, two, two. That's going to give -4. So let's see if we do the same thing. If we could out um negative 44 and you'll end up saying that we won't because F of two times 00. Well, that's still just F of zero. And we just showed that is negative two. So essentially we just showed that two times F of 0, 0 is not equal to F of two times 00. Which would then tell us that is not a home amorphous. Um So implies not homo morph ism Um There is something a little bit more streets or we can do depending on if you have talked about it or not. Um And the thing is is that your identities should get taken to another identity. So for addition, remember this here is the additive identity. Because if we add it to anything, nothing changes. But in just the real numbers, -2 is not the additive identity, it would just be zero. So if you have where an identity does not get taken to another identity, then you can say that it's not a home a morph ISM by that, but uh that's more of a just like theorem that people often have to be taught um or go about proving I should say. But as long as you just show something like this, that would also be a valid way of showing that is not a homo dwarfism.

This question covers topic relating to linear and zebra and the span of uh basis of uh vector space. So you can see here we have the set as a finite and had the route that the the linear combinations of all vectors of S. Is actually there's a set of fee. Okay, so how do we do that? First of all? You refine the uh you can say like we can pick a finite set um the subset of S. And that obviously ideas upset of the vector space V. And because it's the it's an element in sign as actually inside V. So that ai can be rewritten as B J E J. What do you Hj? So E J is um it can be finite or infinite is a basics the basis for V. Okay, so we um so as you can see here um a I can be rewritten as a linear combination of these spaces. Right? And now if we pick any vector, like for example, I pick victor, A and A. Is A and finally linear combination of ai right? So for example, and for I ai right from one, for example, from one to let's say end and maybe uh M. Okay. And so we can re written that some as and for nighttime submission of changing from one to N. B to J. J. And death is just a linear combinations. Uh The finite clinical combination of or the vector E. Uh J. Right? So if you want to write it down precisely, it's just the summation from I from one to M. And information from che from one to end on. For I B J E K. Right? And if you put out the so ehh Okay. So now if you put out each A out put E. J. As you can okay, you can interchange this summation to change this to end and you change it to em. And and for I. B. J. E. J. Right? Okay. So now just uh J equal to one to end of E. J. Um, submission I from 1 to 1 for I beta J. Okay, So this is just a real number, right? So it's a linear combination, so that means I belong to the span of ass. Or so that means the span of S. Is upset of fee.

Hello. Real question. Envisages when that F B and ordered field and X. So I said enough. Okay. It has also given that if X less than zero and why less than that then we need to prove that X. Y greater than access it. So let us get to hear that if access less than zero, this can be written as minus Act should be greater than zero. Okay, now here, if y is less than that so Zach minus Y should be greater than zero. Okay, no, these two have become positive quantities. Some multiplication of two positive quantities should be always positive, should always be positive. So we stretch it as minus X. Which is a positive quantity. Now into that minus Y. Which is again a positive wants to know should be positive. Let us open the bracket minus X. Z bless X. Y should be positive. Let us add except to both the sides will be having X way this is minus exceed all. It is minus except plus X. Y. And we are adding acceptable the sides greater than exit. So these two will become zero. So from here we are getting X. Y greater than X zet. So this is the thing we need to prove. Thank you.

Hello. Real question. Envisages when that F B and ordered field and X. So I said enough. Okay. It has also given that if X less than zero and why less than that then we need to prove that X. Y greater than access it. So let us get to hear that if access less than zero, this can be written as minus Act should be greater than zero. Okay, now here, if y is less than that so Zach minus Y should be greater than zero. Okay, no, these two have become positive quantities. Some multiplication of two positive quantities should be always positive, should always be positive. So we stretch it as minus X. Which is a positive quantity. Now into that minus Y. Which is again a positive wants to know should be positive. Let us open the bracket minus X. Z bless X. Y should be positive. Let us add except to both the sides will be having X way this is minus exceed all. It is minus except plus X. Y. And we are adding acceptable the sides greater than exit. So these two will become zero. So from here we are getting X. Y greater than X zet. So this is the thing we need to prove. Thank you.


Similar Solved Questions

5 answers
ART iiTitration of Equilibrium MixturesTube A Tube B Tube € LMM LCM LM JAOnl SmL Z0mL QQ4l_mL OmL LOn 38L]%nConcentration of NaOH (MNaoh) M Final buret reading; mL Initial buret reading, mL Volume of NaOH used (VNaon), mL Moles of NaOH (= VNaonMNaon) Molarity of HCI Moles of HCIMoles of total acid (= VNOHMNaOh)
ART ii Titration of Equilibrium Mixtures Tube A Tube B Tube € LMM LCM LM JAOnl SmL Z0mL QQ4l_mL OmL LOn 38L]%n Concentration of NaOH (MNaoh) M Final buret reading; mL Initial buret reading, mL Volume of NaOH used (VNaon), mL Moles of NaOH (= VNaonMNaon) Molarity of HCI Moles of HCI Moles of to...
5 answers
If observations in the sample are independent, and the sample is large enough, the null/sampling distribution can be normally distributed: TrueFalse
If observations in the sample are independent, and the sample is large enough, the null/sampling distribution can be normally distributed: True False...
5 answers
Consider the graph of f (z) belowAnswer the following:1 Find the average rate of change over [-1,1] [Select ]2 Identify an interval for which the average rate of change is 0 Select |
Consider the graph of f (z) below Answer the following: 1 Find the average rate of change over [-1,1] [Select ] 2 Identify an interval for which the average rate of change is 0 Select |...
5 answers
Restaurant located in Muscat; 2520 meals should be daily prepared: 1,500 meals in the moming and the remaining in the afternoon. But; In one day only 1320 were sold. Determine the ratios of morning meals; aftemnoon meals then sold and non-sold meals in that day and reduce them;
restaurant located in Muscat; 2520 meals should be daily prepared: 1,500 meals in the moming and the remaining in the afternoon. But; In one day only 1320 were sold. Determine the ratios of morning meals; aftemnoon meals then sold and non-sold meals in that day and reduce them;...
4 answers
BrPzOs MeOH 75% MeoOMeOM HOMeoOHOHHz Pd(OH)2 12,X =Br %66 'HOaw 13,X-h11
Br PzOs MeOH 75% Meo OMe OM HO Meo OH OH Hz Pd(OH)2 12,X =Br %66 'HOaw 13,X-h 11...
5 answers
Score: 0 of 1 pt32 of 34 (31 complete)HW Score: 85.29%, 29 0f 346.6.93Question Help(a) I f(X) = 7" and g(x) graph and on the sanie Cartesian plane (D) Find Ihe poini(s) of intersecilon 0f Ihe graphs Of ( and Dy Solving f(x) = g(x) Label any intersecilon points on the graph drawn pan (a (c) Based on Ihe graph solve I(*) > g(X)(2) Choose Ine correci graph below
Score: 0 of 1 pt 32 of 34 (31 complete) HW Score: 85.29%, 29 0f 34 6.6.93 Question Help (a) I f(X) = 7" and g(x) graph and on the sanie Cartesian plane (D) Find Ihe poini(s) of intersecilon 0f Ihe graphs Of ( and Dy Solving f(x) = g(x) Label any intersecilon points on the graph drawn pan (a (c...
1 answers
Identify a function that has the given characteristics. Then sketch the function. $$ \begin{array}{l}{f(-2)=f(4)=0 ; f^{\prime}(1)=0, f^{\prime}(x)<0} \\ {\text { for } x<1 ; f^{\prime}(x)>0 \text { for } x>1}\end{array} $$
Identify a function that has the given characteristics. Then sketch the function. $$ \begin{array}{l}{f(-2)=f(4)=0 ; f^{\prime}(1)=0, f^{\prime}(x)<0} \\ {\text { for } x<1 ; f^{\prime}(x)>0 \text { for } x>1}\end{array} $$...
5 answers
A 2.5kg package rests on the floor of a truck which acceleratesat 1.5m/s. What must the coefficient of friction be so that thepackage does not slide off of the truck?
A 2.5kg package rests on the floor of a truck which accelerates at 1.5m/s. What must the coefficient of friction be so that the package does not slide off of the truck?...
5 answers
Let y be the solution of the equation y 6xyl emaxlsatisfying the condition y (0) = 0.Let f (x) = e6xy (x) Find the value of the functionat X2
Let y be the solution of the equation y 6xyl emaxl satisfying the condition y (0) = 0. Let f (x) = e6xy (x) Find the value of the function at X 2...
4 answers
At A"AaAo H A ~ ? ~ AIl 13 4 B16 Comparative analysis-Fill in table with tve negative Sign(s): Chromosome Chromosome Centrioles Structure Spindle movement Interphase fibers Prophase Prometaphase Metaphase Anaphase TelophaseNudeus?Nucleolus18. Wnat are differences between mitosis and meiosis?19. What are the stages of Melosis and Melosis Il?20. How many daughter cells are formed at the end of melosis and Melosis 4
At A" Aa Ao H A ~ ? ~ A Il 13 4 B 16 Comparative analysis-Fill in table with tve negative Sign(s): Chromosome Chromosome Centrioles Structure Spindle movement Interphase fibers Prophase Prometaphase Metaphase Anaphase Telophase Nudeus?Nucleolus 18. Wnat are differences between mitosis and meio...
5 answers
Show that f(x)= (x-3)(x+1)^2 on [-1,3] satisfies the hypotheses of Rolle's Theorem then, find all c satisfy the conclusion PLEASE WRITE LEGIBLE. THANKS IN ADVANCE
Show that f(x)= (x-3)(x+1)^2 on [-1,3] satisfies the hypotheses of Rolle's Theorem then, find all c satisfy the conclusion PLEASE WRITE LEGIBLE. THANKS IN ADVANCE...
5 answers
The critlcal value(s s(are) (Round t0 Ihree dec mal placas as needed Use comma (0 saparate answons as naaded |Ihe null hypothesis. Thufo` papulalion vanance for Countyavidoncu Ihat the popu ation variance for CountyUso technology to find Iho p-valuu and Intorpret the rusullThe p-valug (Round lo Ihroa doclmal places neadud )Suco Ino p-Vulua populalon varlance Ior CounlyIha null hypolnonia. Mnero Countyuviduncu coricludo thut Uie
The critlcal value(s s(are) (Round t0 Ihree dec mal placas as needed Use comma (0 saparate answons as naaded | Ihe null hypothesis. Thufo` papulalion vanance for County avidoncu Ihat the popu ation variance for County Uso technology to find Iho p-valuu and Intorpret the rusull The p-valug (Round lo...
5 answers
(e)Draw a Lewis structure of the conjugate acid of NOz - (Ka of the conjugate acid = 4.6 10-4). Calculate the pH of 1.00 L of a 0.100 M solution of NO2 _ Calculate the pH of 1.00 L of a 0.100 M solution of NOz - to which 1 g of H2SO4 has been added. (The first ionization of H2SO4 is complete in aqueous solution; HSO4 has Ka = 1.2 x 10-2,))(g
(e) Draw a Lewis structure of the conjugate acid of NOz - (Ka of the conjugate acid = 4.6 10-4). Calculate the pH of 1.00 L of a 0.100 M solution of NO2 _ Calculate the pH of 1.00 L of a 0.100 M solution of NOz - to which 1 g of H2SO4 has been added. (The first ionization of H2SO4 is complete in aqu...
5 answers
3 1 Jote: Overall UL You can ecoided Scoit a partia "9688 problom 3 tlmes, IlWrite all Pant 2 WJ Previous comnenpaints Howi many cornur Points 8 Problem Nomework following doos 1 space provided system of inequalities: 2: System below; Mave Il 1 2 Eng MuItIp 0 cainer I^ Iv M polnts ,each paleproedetdiou 1
3 1 Jote: Overall UL You can ecoided Scoit a partia "9688 problom 3 tlmes, Il Write all Pant 2 WJ Previous comnenpaints Howi many cornur Points 8 Problem Nomework following doos 1 space provided system of inequalities: 2: System below; Mave Il 1 2 Eng MuItIp 0 cainer I^ Iv M polnts , each palep...
5 answers
14. (ISpts) Given the compler number -3 _ 3i Sketch the graph _ first to help you determine the argument(6) Write this ccmplex nunber in trigorometric for; using rdiens.
14. (ISpts) Given the compler number -3 _ 3i Sketch the graph _ first to help you determine the argument (6) Write this ccmplex nunber in trigorometric for; using rdiens....
4 answers
F(x,Y) = (2xyl + 8Ji + (3xy + 2eZy)jis conservative by finding potential function for F, and use to computedr, where C is the cunve given byr(t) = (2 sin" t)i + 2 sin"(0)ifor 0 <1 < r/2.flx,Y) =(Answers should not have "Ckc dr =
F(x,Y) = (2xyl + 8Ji + (3xy + 2eZy)j is conservative by finding potential function for F, and use to compute dr, where C is the cunve given by r(t) = (2 sin" t)i + 2 sin"(0)i for 0 <1 < r/2. flx,Y) = (Answers should not have "C kc dr =...

-- 0.023119--