Question

i will medal and fan.

Answer 1

go for it :)

Answer 2

What's your question @GlLojei ?

Answer 3

ya, you have a question?

Answer 4

@GlL.ojei

Answer 5

yes.

Answer 6

show that the set \[s={x + y \sqrt[3]{3} + z \sqrt[3]{9} ;x,y,z \in Q}\] is a ring with respect to addition and multiplication on R

Answer 7

wow what class is this from?

Answer 8

This looks like abstract algebra

Answer 9

undergrad/grad level course

Answer 10

just out of curiosity, have you been solving problems similar to this before and this one is particularly tricky or do you need to learn how to solve this problem so you can have the steps to solve others like it?

Answer 11

Looks like they are offline, so we may just never know... :)

Answer 12

ahh so sad

Answer 13

Rings are actually not too bad, they are just tedious requiring you to show that the given formula satisfies associative, commutative, and other properties.

Answer 14

ok. i want to know how to solve questions like this, and please help in solving this. i really want to learn it. please help with the solution,,. and it is an abstract algebra

Answer 15

OK, rings satisfy the following properties:
1) They are a commutative (Abelian) group under addition.
2) Multiplication is associative and has a multiplicative identity.
3) We have left and right distribution of multiplication over addition.

Answer 16

Before we go on, are you familiar with the terms that I am using? "group" "Abelian" "identity" and so on?

Answer 17

yeah. abelian is just another name for commutativity under groups or rings

Answer 18

Alright, excellent. Now, let's roll up our sleeves and begin by working out that the given Ring is an Abelian group under addition.

Answer 19

We have several properties of Abelian groups:
1) They are closed.
2) They are associative.
3) They have an identity.
4) They have inverses.
5) They are commutative.

Answer 20

We are told that the ring uses addition on R, which is good news. This means that we can say that 2) is satisfied automatically because addition on R is associative by definition.

Answer 21

@jtvatsim please continue with the solution .my computer battery is low. will come online with my phone but might not be able to post. please bear with my. i will be watching the solution with my phone. please help complete the solution. will write you or post when am back online with my computer

Answer 22

use a1-a2 for both additive and additive inverse, save time!! :)

Answer 23

yup

Answer 24

let \(s_1=x_1 + y_1 \sqrt[3]{3} + z_1 \sqrt[3]{9} \\s_2=x_2 + y_2 \sqrt[3]{3} + z_2 \sqrt[3]{9} \) in s
take \(s_1-s_2\) you will have a form of s, just conclude : then s is closed under additive and additive inverse.

Answer 25

for multiplicative, multiple them together, after combine like term, you have the product has the form of s, then make conclusion again.

Answer 26

dat sit

Answer 27

However, in your paper, you MUST write out details, like "since x1,x2 in Q and we know that Q is a ring, hence (x1-x2) in Q also, let say x1-x2 = X).... similar for y1 -y2 and z1-z2
to make sure that the terms of s1-s2 satisfy the condition, and get the full credit.hehehe... abstract algebra!!! my nightmare

Answer 28

Thanks a lot!! please, give him/her details. I know the feeling of the beginner.

Answer 29

@GIL.ojei got it or need more??

Answer 30

Full details:

Answer 31

W.........OOOOOOOOOAAAAAAAAAAAHHHHHHHHHH.....impressive

Answer 32

i love this. really brilliant. thanks