If r is a commutative ring, show that the characteristic of r[x] is the same as the characteristic of r.

Answers

Answer 1

If R is a commutative ring, then, the characteristic of R{x} is the same as the characteristic of R.

What is a commutative ring?

In ring theory, a commutative ring can be defined as a ring in which the multiplication operation is considered to be commutative in nature.

Let R represent a commutative ring with characteristic K.

Therefore, we have: Kr = 0∀r ∈ R.

Assuming f(x) ∈ R(x)

Then, f(x) would be given by: [tex]a_n x^n + a_{n-1} x^{n-1} +.......+ a_1 x^1 +a_0=0[/tex]

Also, in some instances ai ∈ R and n ∈ N.

Finally, we would have:

[tex]Kf(x) = K(a_n x^n + a_{n-1} x^{n-1} +.......+ a_1 x^1 +a_0)\\\\Kf(x) = Ka_n x^n + Ka_{n-1} x^{n-1} +.......+ Ka_1 x^1 +Ka_0\\\\Kf(x) = 0\times x^n + 0\times x +.......+ 0[/tex]

Kf(x) = 0.

Therefore, we can infer and logically deduce that if R is a commutative ring, then, the characteristic of R{x} is the same as the characteristic of R.

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Related Questions

When balancing equations, a ____ can be placed to the left of a formula of a substance to make the equations balanced

Answers

Answer:

I think the answer would be a coefficient

Explanation:

Hope this helps:)....if not then sorry for wasting you time and may God bless you:)

Revise sentence 3 by adding the correct verb form. the underground railroad a network of hiding places.

Answers

Answer:

Was

Explanation:

The correct form of the verb used in the above sentence is "Was".

What is Verb?

A verb is a word that, depending on its syntax, often denotes an event, an action, or a state of being. The infinitive, with or without the verb too, is the fundamental form in the typical description of English. Verbs are inflected in several languages to represent tense, aspect, mood, and voice.

The complete sentence is - "The underground railroad was a network of hiding places." Was is the past tense suggestive form of be, which means "to exist or live," and it may be used in both the first and third person singular (I, he/she, it). When discussing reality and established facts, you utilize the past indicative.

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What is the overall purpose of queen elizabeth's address

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Answer:

the passage from her address to the troops at Tilbury, Queen Elizabeth I makes appeals to ethos and pathos. To encourage her troops, she explains that she supports and relies on them, and she tries to stir their feelings of loyalty toward her.

The security classification guide (scg) states: (u) mission (c) personnel (u) location (s) mission and location the new document states: bravo company will be departing on 1 september for exercise peace keeper. the exercise will be a joint effort with the army and the marine corps. both forces will meet in qatar to complete the exercise. which concept was used to determine the derivative classification of the new document?

Answers

The security classification guide (SCG) is a comprehensive guideline. The new document will be classified as 'S' and use the secret concept to define the derivative classification.

What is Security Classification Guide?

Security Classification Guide (SCG) is the complete collection of the system, specific program, and weapon system. It is used to make classified documents.

The document will be classified 'S' as it states the mission will be jointly done by corps and army and the location will be delivered on the day of the departure.

Therefore, the new document will be classified as 'S.'

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Evidence such as spectra from distant stars suggests that the universe is... what?

Answers

Answer - the universe is expanding.

Dr. smithe is designing a study to investigate whether aggressive preschoolers are likely to become aggressive adolescents. dr. smithe should use

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According to the research, Dr. Smithe should use naturalistic observation to investigate aggressive preschoolers.

What are naturalistic observations?

It is a method used in research and whose basis is to observe the subjects in the place where they live, without the researcher being disturbed in the least.

This allows detailing the behavior of the subjects that we record is what, in fact, the behavior that should happen in their habitat, under those conditions, obtaining real results, without any type of alteration.

Therefore, we can conclude that according to the research, Dr. Smithe should use naturalistic observation to investigate aggressive preschoolers.

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Which will have cells? (Select all that apply.)
Animals monkeys plants or food

Answers

[tex] \boxed{ \frak{Answer}}[/tex]

As we that all living organisms have cells. Whether it is a plant or an animal. Cell is a fundamental unit of life. For an organism to live cell is mandatory.

And food is non living.

So,

Monkeys Plants

Will have cells

[tex] \boxed{ \frak{Be \: Brainly}}[/tex]

Answer: animals monkey

Explanation:

Which word is an antonym for the word gaunt brainly

Answers

Answer:

haggard

Explanation:

i got it right

Last year, about 2,400 people participated in a local event. this year, about 3,200 people participated. what was the approximate percent increase in participation? 25% 33% 50% 75%

Answers

The approximate percent increase in participation in a local event is 33%.

What is percentage increase?

The increase in the involvement of an activity compared to the previous data expressed in percentage.

Given  is the 2,400 people participated in a local event. This year, about 3,200 people participated.

Then percentage increase will be

% increase = [(3200 - 2400) / 2400 ]x 100

                 = 33.33%

Thus, the approximate percent increase in participation in a local event is 33%.

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Of the three characters in this episode, which had never been to washington d.c. before? summer jake dr. sabato

Answers

Answer:

jake

Explanation:

Answer:

1. jake

2. five minutes

3. the agency

Explanation:

Of 1000 students surveyed, 490 were boys and 320 had bicycles. of those who had bicycles, 130 were girls. what is the probability that a student surveyed was either a boy or had a bicycle?

Answers

By definition of probability, the probability that a student surveyed was either a boy or had a bicycle is 62%.

Definition of Probabitity

Probability is the greater or lesser possibility that a certain event will occur.

In other words, the probability is the possibility that a phenomenon or an event will happen, given certain circumstances. It is expressed as a percentage.

Union of events

The union of events, AUB, is the event formed by all the elements of A and B. That is, the event AUB is verified when one of the two, A or B, or both occurs. AUB is read as "A or B".

The probability of the union of two compatible events is calculated as the sum of their probabilities subtracting the probability of their intersection:

P(A∪B)= P(A) + P(B) -P(A∩B)

The intersection of events, A ∩ B, is the event formed by all the elements that are, at the same time, from A and B. That is, the event A ∩ B is verified when A and B occur simultaneously.

Events and probability in this case

In first place, let's define the following events:

A: The event that a student surveyed was a boy.B: The event that  a student surveyed had a bicycle.

Then you know:

P(A)= [tex]\frac{490}{1000}[/tex]=0.49= 49%P(B)= [tex]\frac{320}{1000}[/tex]= 0.32= 32%P(A and B)= P(A∩B)= [tex]\frac{190}{1000}[/tex]= 0.19= 19% [320 students had bicycles. Of those who had bicycles, 130 were girls. Then, 190 were boys.]

In this case, considering the definition of union of eventes, the probability that a student surveyed was either a boy or had a bicycle is calculated as:

P(A∪B)= P(A) + P(B) -P(A∩B)

P(A∪B)= 0.49 + 0.32 -0.19

P(A∪B)= 0.62= 62%

Then, the probability that a student surveyed was either a boy or had a bicycle is 62%.

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Anyone needs help with any assignments

Answers

Answer:

meeeemeeeeeeeeeee if possible

Answer:

I don't need help but thx anyways

Explanation:

2. Now, let's turn our attention to memory strategies. (10 points) a. List two of the memory strategies you have learned. b. Provide specific examples of how you could use each memory strategy

Answers

1- Focus attention
2- Read out loud
3- Avoid cramming
4- Relate new information go thing you already know

In the united states, what is minority status usually based on?

Answers

Answer:National,cultural and religious identityidentityidentity.

Explanation:

To avoid an accident, a driver with a mass of 60 kg applies his brakes and decelerates from 30 m/s to 1 m/s in 2 seconds. how much force was absorbed by his seat belt?

Answers

When brakes are applied, force absorbed by his seat belt was  870 N.

What is force?

The force is an action of push or pull in order to move or stop the body.

Given is the mass m=60 kg, velocity changes from v1 =30m/s to v2= 1m/s in time t=2s.

The acceleration is the time rate of change of velocity.

a = v1 -v2 /t

a =30 -1 /2

a = 14.5 m/s²

According to the Newton's second law of motion, the force absorbed by belt was

F =ma

F =60 x 14.5

F = 870 N.

Thus, force absorbed by his seat belt was 870 N.

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Of the 34 cars in a parking lot, 12 are blue, 8 are red, 4 are white, and 10 are silver. in simplest form, what are the odds in favor that the next car that leaves the lot will be red or white? a. 6:11 b. 6:17 c. 17:6 d. 11:6

Answers

Answer:

see the attachment with the answer

If you turn the microwave on for two minutes, is the rate of popping always the same, or does it change? explain.

Answers

The rate of popping is not always the same because by the end of the two minutes, most of the popcorn is popped so you hear pops at a slower rate than at the beginning when most of the popcorn wasn't popped.

What effect does the modality of this exhibit most likely have on the visitors to the museum?visitors can use the gustatory modality to taste the tomatoes picked by laborers.visitors can use the auditory modality to hear stories about modern enslavement first-hand.visitors can use the tactile modality to feel the weight of the heavy tomato buckets.visitors can use the logic modality to calculate and compare earnings.

Answers

Answer:

Visitors can use the tactile modality to feel the weight of the heavy tomato buckets.

Explanation:

Got it right

The trans-pecos eco-region in the western portion of texas receives less than 15 inches of rainfall in a year and has little soil due to overgrazing by farm animals. what catastrophic event is most likely to cause the most weathering and erosion in the least amount of time in this eco-region?

Answers

Answer:

Explanation:

On\

Which answer best describes the overall structure of the part of the text in which louis pasteur helps joseph meister?

Answers

The best description of overall structure of the part of the text is it explains what causes Pasteur to want to find a cure for rabies.

Who was Louis Pasteur?

Louis Pasteur is a famous French chemist and microbiologist. He is known for discovering the cure for rabies.

He is also known for preventing diseases in silk eggs. Pasteurization is the process of preventing microbes that spoil milk or other beverages. The process is in his name.

Thus, the correct option is A, it explains what causes Pasteur to want to find a cure for rabies.

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What is the main purpose of sustainable agriculture?

Answers

Answer the goal is to reduce the input of external energy and to substitute non-renewable energy sources with renewable sources

The speaker begins the poem by stating "i come back to your youth, my nana" (line 1), but the primary transformation sought overall is

Answers

The speaker which begins the poem has the primary transformation sought overall is recapturing of a sense of life's possibilities.

What is primary transformation?

The transformation in the poem means the transformation of the poet into the most authentic self. Primary transformation tells the meaning of the line of poem which the poet has written in the self authentic manner.

The speaker begins the poem by stating ,

"i come back to your youth, my nana"  

Here, the primary transformation sought overall is recapturing of a sense of life's possibilities.

The speaker which begins the poem has the primary transformation sought overall is recapturing of a sense of life's possibilities.

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Which sample size will produce the widest 95% confidence interval, given a sample proportion of 0.5?

Answers

The sample size that will produce the widest 95% confidence interval, is 50. The correct option is D.

What is the confidence interval of proportion?

The confidence interval of proportion is the sample proportion multiplied by the standard deviation of the sample proportion.

π is the sample proportion, 0.5

z is the critical value, 95%

n is the sampling size,

Thus, the correct option is D, 50.

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By law, manufacturers and distributors are responsible for providing safety data sheets. who else is responsible for providing safety data sheets? select the best option. managers workers importers employers

Answers

By law, the other persons that are responsible for providing safety data sheets (SDSs) are: C. importers.

What is a Safety Data Sheet (SDS)?

A Safety Data Sheet (SDS) can be defined as a text-based document which is developed to provide more information with respect to the chemical and physical properties of an equipment, chemical compound, or substance, based on the following:

FlammabilityTemperatureRadioactivity

By law, the people that are responsible for providing safety data sheets (SDSs) are:

ImportersManufacturersDistributors

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A common motif in the poetry of this unit is death

Answers

Various poets address the element of death in their poems by referring to it as something dark.

How poets address the element of death

Various poets address the element of death in their poems by mentioning something that relates to dark.

Death is used as a motif when it uses a symbol to represent death.

Thus, poets address the element of death in their poems by referring to it as something dark.

The complete question is below:

A common motif in the poetry of this unit is death. How do the various poets address the element of death?

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Paul needs to randomly select one of his four children to go first in a board game. which simulation tools could he use in this situation?

Answers

The simulation tools that could be used by Paul in this situation is :

A spinner divided evenly into 12 sections, with three sections each of four different colors A standard deck of cards. Two coins

Description of the simulator tools

A spinner divided into 12 sections with three sections each having four different colors can be used by calculating the prob of each subsection which simulates the probability of picking one of four children

Probability of each subsection

=   number subsection / number of section

= 3 / 12 =  1/4

Two coins can also simulate the picking of one of four children

The outcome of the two coins = ( HH, HT, TT, TH )

Therefore the probability of picking a H or T

= 1 / 4

Also deck of 52 cards can also simulate the probability of picking one out of four children to go first in a board game.

Hence we can conclude that the simulator tools that can be used are A spinner divided evenly into 12 sections, with three sections each of four different colors A standard deck of cards. and Two coins.

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Golf ball sized basketballs are often packed in cylindrical canisters

Answers

The correct statement about the circumference (C) and the height (h) is gotten as; C > h

How to find the circumference of the cylinder?

The formula for circumference of a cylinder is;

C = 2πr

Now, we are not given the radius of the cylinder but we are told that there are 3 balls that fit perfectly into the cylindrical canister. Each having a diameter of 1.68 inches.

Thus;

Radius; r = d/2 = 1.68/2 = 0.84 inches

Height of cylinder will be;

h = 3d = 3 * 1.68

h = 5.04 inches

Circumference; C = 2π * 0.84 = 5.28 inches. Thus;

C > h

The complete question;

Golf ball size basketballs are often packed in cylindrical canisters. The diagram below shows three balls in a canister. Let C be the circumference of the canister and let H be the height of the canister. Assuming each ball has a diameter of 1.68 inches and the balls fit snugly in the canister, which of the following statements is correct?

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Review the excerpt from tennyson's "lady of shalott, then answer the question. willows whiten, aspens quiver, little breezes dusk and shiver thro' the wave that runs forever by the island in the river flowing down to camelot. four gray walls and four gray towers, overlook a space of flowers, and the silent isle embowers, the lady of shalott. how many feet are used in each of the first 4 lines? feet

Answers

Based on the excerpt from Tennyson's "Lady of Shalott," the number of feet that are used in each of the first 4 lines is 4 feet.

What is a poem?

A poem can be defined as a literary work that's made up of either written words which are carefully chosen and arranged in separate lines, especially based on their sound (diction), rhythm, and meaning.

Based on the excerpt from Tennyson's "Lady of Shalott," we can logically deduce that the number of feet that are used by the poet in each of the first 4 lines is 4 feet.

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In which situation is nuclear power most beneficial? if there is a low concentration of people if there is a nearby supply of fossil fuels if there is an isolated location if there are many other energy resources

Answers

Answer:

if there is an isolated location

Explanation:

took the test

Two football players with mass 80 kg and 120 kg run directly toward each other with speeds of 8 m/s and 10 m/s respectively. if they grab each other as they collide, the combined speed of the two players just after the collision would be

Answers

The combined speed after the collision is -2.8 m/s.

How to get the combined speed?

Here we must have conservation of momentum, where momentum is the product between mass and velocity.

At the beginning, one of the players has a mass of 80kg and a velocity of 8m/s, so its momentum is:

P₁ = 80kg*(8m/s) = 640 kg*m/s

The other player has a mass of 120 kg and runs in the opposite direction at a speed of -10m/s (the negative sign is because of the direction). So its momentum is:

P₂ = 120kg*(-10m/s) = -1200 kg*m/s

The initial total momentum is:

P = P₁ + P₂ = 640 kg*m/s + ( -1200 kg*m/s) = -560 kg*m/s

At the end, the move together as a mass of (80kg + 120kg) = 200kg with a velocity V.

Such that the final total momentum is equal to the initial total momentum.

So we must have:

200kg*V =  -560 kg*m/s

V = (-560 kg*m/s)/200kg = -2.8 m/s.

The combined speed after the collision is -2.8 m/s.

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