# Thread: subring of the integer set

1. ## subring of the integer set

Why isn't $\displaystyle Z_2 = \{0, 1\}$ a subring of the integer set $\displaystyle Z$?

2. Originally Posted by dori1123
Why isn't $\displaystyle Z_2 = \{0, 1\}$ a subring of the integer set $\displaystyle Z$?
Because it is not even a subset !

3. why not? $\displaystyle Z_2$ is contained in $\displaystyle Z$, isn't it?

4. Originally Posted by dori1123
why not? $\displaystyle Z_2$ is contained in $\displaystyle Z$, isn't it?
Ah! But remember it is not really $\displaystyle \{ 0,1\}$ it is really $\displaystyle \{ [0]_2,[1]_2\}$.

Where, $\displaystyle [0]_2 = \{ 0, \pm 2, \pm 4, ... \}$ and $\displaystyle [1]_2 = \{ \pm 1, \pm 3, \pm 5, ... \}$.

Therefore, $\displaystyle \mathbb{Z}_2$ is a group modulo $\displaystyle 2$ under addition.

5. Maybe she has mistaken it with $\displaystyle 2\mathbb{Z}$ which are the even integers and indeed is a subring of the integers, and so are $\displaystyle n\mathbb{Z}$

6. That depends on what is a ring: if you make the difference between rings and pseudo-rings, the only subring of $\displaystyle \mathbb{Z}$ is $\displaystyle \mathbb{Z}$ itself.

7. Please, donīt make things more complicated.

A ring is a ring is a ring....

8. Well "my" definition of a ring $\displaystyle A$ states that $\displaystyle A$ has a multiplicative identity...

9. Originally Posted by dori1123
Why isn't $\displaystyle Z_2 = \{0, 1\}$ a subring of the integer set $\displaystyle Z$?
Another way of putting it is that the addition operation is different. In $\displaystyle \mathbb{Z}$, 1+1=2. But in $\displaystyle \mathbb{Z}_2$, 1+1=0.

Part of the definition of a subring is that it must inherit the algebraic operations of the full ring.

10. I donīt think it was necessary to lower my reputation for my comment; obviously we are all talking about rings with unity, so my comment about the subrings of $\displaystyle \mathbb{Z}$ is correct in that context.
Bringing to the table the issue of pseudo-rings (or rngs) had nothing to do with the question, must insist...

11. Inti, maybe I don't understand, but what's the unity in $\displaystyle n\mathbb{Z}$?

ps. I didn't lower your reputation