# Thread: 3 exercises-groups,rings,ideals

1. ## 3 exercises-groups,rings,ideals

1)prove that every finite subgroup of Q/Z is cyclic.

2. Originally Posted by tom007
1)prove that every finite subgroup of Q/Z is cyclic.
2)let R={a/b; a,b ∈ Z and b is not divisable by 11} Show that R is factorization domain. How many prime elements are in R up to associates (Elements a,b are said to be associates if a | b and b | a. ) ? How do ideals look like in R?
3)a)Find all invertible elements in Z/10Z. Describe a group of invertible elements up to isomorphism. Find all classes associates in Z/10Z.
b)Find all maximal ideals of Z/10Z
c)Find all prime ideals of Z/10Z
d)Find all irreducible elements in Z/10Z
e)Find all prime elements in Z/10Z

pls help

If you want help then first show what've you done so far in each exercise, otherwise it's like asking others to give you the whole answer.

Tonio

3. .

4. 1) G={m1/n1 + Z,..., mk/nk + Z} finite simple of Q/Z where gcd(mi,ni)=1, set n=lcm(n1,...nk)
i don't know how to show that G is <1/n + Z>

2) invertible :b/a 11 doesnt divide 1 irred: 11k/b

3) Z/10Z ={(1),...(9)}
(2)=(4)
inver: (1)^(-1)=1, (2)^(-1)=8, (3)^(-1)=7, (4)^(-1)=4,(5)^(-1)=5,(6)^(-1)=6,(7)^(-1)=3,(8)^(-1)=2,(9)^(-1)=9

5. Tom007's solution is incorect.
you need to understand what operation is invovled in Z/10Z here.

6. .

7. is there any1 who could help me with #3

8. Originally Posted by tom007
is there any1 who could help me with #3

An element $a\in\mathbb{Z}\slash n\mathbb{Z}$ is invertible iff $(a,n)=1$

Tonio

9. tnx alot Tonio..i'll try now sth with this...(sun)
do u know #1?

sorry for bothering u.. i don't have much time, and have to do many things.. so, any help is welcome
i really appreciate ur help Tonio!

10. Originally Posted by tom007
tnx alot Tonio..i'll try now sth with this...(sun)
do u know #1?

sorry for bothering u.. i don't have much time, and have to do many things.. so, any help is welcome
i really appreciate ur help Tonio!

Let $G\le\mathbb{Q}\slash\mathbb{Z}$ be a finite group. Let us agree to write elements in $\mathbb{Q}\slash\mathbb{Z}$ as $\frac{\alpha}{\beta}+\mathbb{Z}\,,\,\,\alpha\,,\,\ beta\in\mathbb{Z}\,,\,\,\beta\neq 0\,,\,\,(\alpha,\beta)=1$ .

$\mbox{Let }\frac{a}{n}+\mathbb{Z}\in G\mbox{ be such that }\forall\,\,\frac{\alpha}{\beta}+\mathbb{Z}\,,\,\, n\geq \beta$. Now prove the following:

1) $\exists\,\,x\,,\,y\in\mathbb{Z}\,\,\,s.t.\,\,\,ax+ ny=1$

2) From the above it follows that $\frac{1}{n}+\mathbb{Z}\in G$

3) If there's an element $\frac{r}{s}+\mathbb{Z}\in G$ s.t. $\frac{r}{s}+\mathbb{Z}\notin \left<\frac{1}{n}+\mathbb{Z}\right>$ , then we get a contradiction to the maximality of n

4) It follows that $G=\left<\frac{1}{n}+\mathbb{Z}\right>$

Tonio

Ps. BTW, from the above it almost follows, adding some minor details, that $\mathbb{Q}\slash\mathbb{Z}$ has a unique (cyclic, of course) subgroup of order $n$ for every $n\in\mathbb{N}$

11. thank you very much Tonio!!!!