1. Find ideals

Hi!

I have problem with this exercice

Find three ideals $(a)$ in $\in{\mathbb{Z}}$ with thr property that $(24)\subsetneq (a)$ ( $\subsetneq$ means is a proper subsets of)

- Definition

An ideal in a commutative ring $R$ is a subset $I$ of $R$ such that

$a)$ $0\in{I}$
$b)$ $a,b \in{I} \Rightarrow{a+b \in{I}}$
$c)$ If $a\in{I}$, $r\in{A}$ then $ra \in{I}$

I think of $(2)$ $(4)$ and $(6)$ but are not ideal since $(b)$fails

2. Re: Find ideals

First, you mean " $r\in R$" rather than " $r\in A$", don't you? (2), (4), and (6) are the sets of all multiples of 2, 4, and 6, respectively, right? Then it certainly is true that "if a and b in I then a+ b in I". if a and b are multiples of r, then a= mr, and b= nr so a+ b= mr+ nr= (m+ n)r.

3. Re: Find ideals

Means $r \in R$ , Second for example $(4)=\{mr:\,r\in\mathbb{Z}\}=\{\dots,-12,-8,-4,0,4,8,12,\dots\}.$ but I do not understand this condition $(24)\subsetneq (a)$

The ideals (2) (4) and (6) satisfy the condition $(24)\subsetneq (a)$ ?

4. Re: Find ideals

Originally Posted by cristianoceli
$(24)\subsetneq (a)$ ( $\subsetneq$ means is a proper subsets of)
Now you post:
Originally Posted by cristianoceli
Means $r \in R$ , Second for example $(4)=\{mr:\,r\in\mathbb{Z}\}=\{\dots,-12,-8,-4,0,4,8,12,\dots\}.$ but I do not understand this condition $(24)\subsetneq (a)$
The ideals (2) (4) and (6) satisfy the condition $(24)\subsetneq (a)$ ?
So we are rightly confused as to what you do understand. Do you see why that is?

Surely you are not saying that you do not understand the notation: $B\subsetneq A~?$
If you do not understand the notation it means: $[B\subset A] \wedge[\exists x\in A$ such that $x\notin B]$

If you know what $(24)$ is then is it true that $(24)\subsetneq (4)~?$

5. Re: Find ideals

Originally Posted by cristianoceli
Means $r \in R$ , Second for example $(4)=\{mr:\,r\in\mathbb{Z}\}=\{\dots,-12,-8,-4,0,4,8,12,\dots\}.$
You mean $(4)= \{4r: r\in\mathbb{Z}\}$. But yes, that is $\{\dots,-12,-8,-4,0,4,8,12,\dots\}$, all multiples of 4. Those dots of course, mean that it continues in that way- it also contains -24= 4(-6), 24= 4(6), -48= 4(-12), 48= 4(12), etc.

but I do not understand this condition $(24)\subsetneq (a)$
$(24)= \{\dots, -48, -24, 24, 48, \dots\}$ consists of all multiples of 24 which are all multiples of 4 because 24 itself is a multiple of 4.

Similarly, $(6)= \{\dots, -30, -24, -18, -12, -6, 0, 6, 12, 18, 24, 30, \dots\}$ which, again, contains all multiples of 24 because 24 is a multiple of 6.

The ideals (2) (4) and (6) satisfy the condition $(24)\subsetneq (a)$ ?

6. Re: Find ideals

Originally Posted by Plato

If you know what $(24)$ is then is it true that $(24)\subsetneq (4)~?$
Yes, $[(24)\subset (4)]$ y for example $8 \in (4)$ but $8 \not\in{(24)}$. Therefore $[\exists x\in (4)$ such that $x\notin (24)$

7. Re: Find ideals

You have it backwards. (24) is a subset of (4). What your example shows is that (4) is not a subset of (24).