Ideals in rings

[lillucy31]

Hi, I am really struggling with a quetion about ideals on my rings, polynomials and fields paper. The question is:

Show that the set

I={a+b(-5)^1/2:a,b in Z and a-b is even}

Is an ideal in Z[-5]^1/2

I do not have a lot of knowledge on Ideals soa simple explanation would be great. Would appreciate any help on Ideals at all.

 

[Swlabr]

Hi, I am really struggling with a quetion about ideals on my rings, polynomials and fields paper. The question is:

Show that the set

I={a+b(-5)^1/2:a,b in Z and a-b is even}

Is an ideal in Z[-5]^1/2

I do not have a lot of knowledge on Ideals soa simple explanation would be great. Would appreciate any help on Ideals at all.

Firstly, this is an algebra question so you would be more likely to get a quick reply if you had posted there...but someone will move this thread soon I am sure!

Now, an ideal \(\displaystyle I\) of a ring \(\displaystyle R\) is a set which is closed under addition and under multiplication by and element of the ring. So, \(\displaystyle a+b \in I\) for all \(\displaystyle a, b \in I\), \(\displaystyle ar \in I\) and \(\displaystyle ra \in I\) for all \(\displaystyle r \in R\) (although your ring is commutative so you do not need to show both \(\displaystyle ra\) and \(\displaystyle ar\) are in \(\displaystyle I\) here).

So, you need to check these two things here.

Let \(\displaystyle a_1 + b_1 \sqrt{-5}\) and \(\displaystyle a_2 + b_2 \sqrt{-5}\) be in the ideal and let \(\displaystyle x + y \sqrt{-5}\) be an arbitrary element of the ring.

Does \(\displaystyle a_1 + a_2 + (b_1 + b_2)\sqrt{-5}\) satisfy the condition? What about \(\displaystyle (a_1 + b_1 \sqrt{-5})(x+y\sqrt{-5})\), once you have multiplied it out?

 

[lillucy31]

Firstly thanks for the quick reply despite it being in the wrong section, very new at this whole thing.
does satisfy the condition because it's in the same form as the original set?
I've multiplied out the to get a_1(x+y\sqrt{-5}) + b_1(x\sqrt{-5} - 5y) but I'm not sure if this satisfies the condition, or whether I've written it in the write form. Could you possibly explain why it does satisfy the conditions?

Thanks again. (Nod)

 

[Swlabr]

Firstly thanks for the quick reply despite it being in the wrong section, very new at this whole thing.
does satisfy the condition because it's in the same form as the original set?
I've multiplied out the to get a_1(x+y\sqrt{-5}) + b_1(x\sqrt{-5} - 5y) but I'm not sure if this satisfies the condition, or whether I've written it in the write form. Could you possibly explain why it does satisfy the conditions?

Thanks again. (Nod)

Put the maths stuff in [math ] [/math ] tags, then it will be presented nicely.

What does it mean for an element to be in this ideal?

 

[lillucy31]

Im really sorry I have no idea, the notes I have from my lectures aren't useful, so I don't know much.

 

[Swlabr]

Im really sorry I have no idea, the notes I have from my lectures aren't useful, so I don't know much.

You have been given a set under a condition. You need to show that if you take two elements each of which satisfy this condition then their sum will satisfy the condition. Further, if you take an element of this set and an arbitrary element from the ring then their product will satisfy this condition.

So, for example, \(\displaystyle (a_1 + a_2) + (b_1 + b_2)\sqrt{-5} = \alpha + \beta \sqrt{-5}\) where \(\displaystyle \alpha = a_1 + a_2\) and \(\displaystyle \beta = b_1 + b_2\). Is \(\displaystyle \alpha - \beta\) even?

For the multiplying a ring element with an ideal element, multiply them together, then collect the \(\displaystyle \sqrt{-5}\) terms. Then you have to work out if the \(\displaystyle \sqrt{-5}\) coefficient minus the other stuff gives you an even number.

In both cases you need to use the assumption that you are in the ideal so the elements satisfy the condition.

 

[lillucy31]

okay. After having collected the \(\displaystyle \sqrt{-5}\) I got \(\displaystyle \sqrt{-5}(a_1y+b_1x) -a_1x - 5b_1y\) I'm not sure how work out whether it gives me an even number or not.

 

[Swlabr]

okay. After having collected the \(\displaystyle \sqrt{-5}\) I got \(\displaystyle \sqrt{-5}(a_1y+b_1x) -a_1x - 5b_1y\) I'm not sure how work out whether it gives me an even number or not.

Plug that into your condition and what do you get?