1. ## classes

Say you want to show that $(3) \subseteq I \subseteq \mathbb{Z}$ then $I = (3)$ or $I = \mathbb{Z}$. Well (3) is principal. So it is maximal. Thus the result follows? Are there any other ways of showing this?

2. Originally Posted by Sampras
Say you want to show that $(3) \subseteq I \subseteq \mathbb{Z}$ then $I = (3)$ or $I = \mathbb{Z}$. Well (3) is principal. So it is maximal. Thus the result follows? Are there any other ways of showing this?
Supose (3) < I ==> there's a non-multiple of 3 in I, say x ==> since 3 is prime, (x,3) = 1 ==> there exist integers m, n s.t. 3m + xn = 1.
But 3m in (3) < I and xn in I ==> 1 = 3m + xn in I ==> I = Z and we're done.

Tonio

3. Is (3) a coset?

4. Originally Posted by Sampras
Is (3) a coset?

(3) is exactly what you said it is: the principal ideal generated by 3!!

Tonio

5. Then why are you saying: "(3) < Z?" Why not (3) subseteq Z.

6. Originally Posted by Sampras
Then why are you saying: "(3) < Z?" Why not (3) subseteq Z.

Errr...because that's the standard notation for ideals of rings and or subgroups of groups: I is an ideal of ring R can be written I < R...

Tonio