Similar question here Let $R$ be the set of all integers with alternative ring operations defined below. Show that $\Bbb Z$ is isomorphic to $R$. The difference is that in attempting to answer my own problem, I can't.

For any integers $a,b$, define $a\oplus b=a + b - 1$ and $a\odot b=a + b - ab.$ Let $R$ be the ring of integers with these alternative operations. Show that $\Bbb Z$ is isomorphic to $R$.

Let $f: \mathbb{Z} \to R$.

Suppose we have $0, 1, a \in \mathbb{R}$.

Then we have $f(a\otimes 1) = a + 1 - a = 1$.

For $f(a\oplus 1) = a + 1 - 1 = a$.

For $f(a \otimes 0) = a + 0 = a$.

Thus, the multiplicative identity is $0$ and the additive identity is $1$.

I DO NOT KNOW HOW TO DEDUCE THAT THE FUNCTION IS $f(x) = -x + 1$. I need serious help in that. Isn't the identity for multiplication in the second ring "0" where it's 1 in the integers, thus $f(e_G) \neq e_H$ so it isn't a homomorphism?

Now, we check $f(ab) = f(a) \otimes f(b). f(ab) = (-ab + 1); f(a)f(b) = (-a + 1)(-b + 1)$ which is obviously not homomorphic. In a homomorphism, $f(e_\mathbb{Z}) = e_R$. What did I do wrong??

  • $\begingroup$ Your third line is weird on the light of your first two: "This is a..." What is a homomorphism ? $\endgroup$ – DonAntonio Oct 24 '13 at 12:13
  • $\begingroup$ Duplicate of your own question? math.stackexchange.com/questions/535587/… $\endgroup$ – lhf Oct 24 '13 at 12:14
  • $\begingroup$ @lhf: it's not a duplicate. $\endgroup$ – Don Larynx Oct 24 '13 at 12:21
  • $\begingroup$ @DonAntonio: Fixed. $\endgroup$ – Don Larynx Oct 24 '13 at 12:22
  • $\begingroup$ Ok, @DonLarynx...but still it looks odd: you're trying to define a function not on single elements but on their sum/multiplication $\;\otimes\;$ ...how come?! $\endgroup$ – DonAntonio Oct 24 '13 at 12:26

It is important to keep in mind where the different elements are. If $f:\Bbb Z \to R$, then there is no sense in applying $\oplus$ and $\odot$ to the arguments of $f$. Writing something like $f(a\odot b)$ is not very productive, since $a, b$ are considered to be elements of $\Bbb Z$, and not elements of $R$ where these operations makes sense. Rather, you want to apply those operations to the resulting elements, like so: $f(a)\odot f(b)$.

The fact that they are both rings defined on the set of integers (showing that $R$ is in fact a ring is important as well) does make it difficult to keep $\Bbb Z$ and $R$ apart, but it is necessary to do so in order to understand what is going on. If you want to show to show that $f$ is a homomorphism, you want to show the following:

  1. $f(a + b) = f(a)\oplus f(b)$
  2. $f(ab) = f(a)\odot f(b)$

Remember, if $a, b\in \Bbb Z$, then $\oplus$ and $\odot$ have no business being near them. Similarily, if $c, d \in R$, then you need to be very careful about $+$ and $\cdot$, and remember that they are only a means for calculating $\oplus$ and $\odot$, not "real" operations.

I will do point 1 above, I hope you can follow it to show number 2. $$ f(a + b) = -(a + b) + 1 = -a -b + 1 = (-a + 1) + (-b + 1) - 1 \\= f(a) + f(b) - 1= f(a) \oplus f(b) $$

Here is a small addendum on how one might deduce $f$. First of all, we need to identify the additive and multiplicative identities $0_R$ and $1_R$. That can be done by solving the equations $a\oplus 0_R = a$ and $b \odot 1_R = b$. This will result in $1_R = 0$ and $0_R = 1$.

Now, assume there is a (non-trivial) homomorphism $f$. We will use $f(1) = 0$ to deduce what $f(a)$ must be for a general $a$ if it does exist. We can start by deducing $f(2)$: $$ f(2) = f(1+1) = f(1)\oplus f(1) = 0\oplus 0 = 0 + 0 - 1 = -1 $$ Now $f(3)$ and $f(4)$ are calculated similarily to be $-2$ and $-3$, and a pattern seems to emerge; the definition $f(a) = -a + 1$ fits so far. To show that this actually is a homomorphism, we prove the two steps above, and since any homomorphism from $\Bbb Z$ is completely determined by what its value on $1$ is, this is the only one.

Once we show that it is bijective, we know that it is an isomorphism, and therefore, $\Bbb Z$ and $R$ are isomorphic.

  • $\begingroup$ Just adding, that OP should also see which are the neutral elements and inverse for the "sum" in $R$ and check that $f(0) = 0_R$, $f(1) = 1_R$ $\endgroup$ – bernatguillen Oct 24 '13 at 12:38
  • $\begingroup$ I checked for that @Duronman; $f(0 + 0) = 0 - 0 - 1 = -1 \neq f(0) \oplus f(0)$? @duronman $\endgroup$ – Don Larynx Oct 24 '13 at 12:40
  • $\begingroup$ @DonLarynx We have $$ f(0 + 0) = -(0 + 0) + 1 = 1\\ f(0)\oplus f(0) = (-0 + 1)\oplus(-0 + 1) = 1\oplus 1 = 1+1-1 = 1 $$ and they are equal. $\endgroup$ – Arthur Oct 24 '13 at 12:48
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    $\begingroup$ @DonLarynx See my edit. Finding a candidate for $f$ in this case is more or less a guided trial and error (most of the time it is), but once you have a candidate, checking whether it's a (the) correct one is a matter of applying machinery. $\endgroup$ – Arthur Oct 24 '13 at 13:10
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    $\begingroup$ @DonLarynx In short, you should see that $1$ is the additive identity in the new ring, $0$ is the multiplicative identity in the new ring, and a ring homomorphism needs to preserve identities. So it would have to map $1\mapsto 0$ and $0\mapsto 1$. At this point, one might begin to suspect that $1-x$ might do the trick. $\endgroup$ – rschwieb Oct 24 '13 at 13:47

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