In modular arithmetic, is a residue class a vector space? Does it have other structure, e.g., ring or group structure?

Question

I tried working this out on scratch paper and don't believe that it is a vector space -- a residue class doesn't seem closed under addition.

Am I mistaken?

My guess was that it is a vector space; in particular, that it is a "quotient space", where all the integers that have $0$ residue, modulo some integer $p$, just collapses to the "zero vector" in the quotient space.

Can someone briefly explain whether I am on the right track?

I am a beginner in abstract algebra and elementary number theory, although I have experience in linear algebra, where the topic of quotient space came up very briefly.

Thanks,

Answer 1

I'll give you as big of an answer I can to cover all grounds. A vector space is a module where the ring is actually a field, the most common one by far is $\mathbb{R}$, second is $\mathbb{C}$ and third $\mathbb{Q}$, while it is possible to use finite fields $\mathbb{F}_p$ I have personally never encountered them.

As you are asking about modulo classes it is safe to assume that the supervector space would be $\mathbb{R}$, as an initial arguement point, here modulo classes do not form a quotient vector space, why? Simple, $n\mathbb{Z}$ does not form a subspace because multiplication by any real number will take it out of the integers.

If you meant just a single modulo class $a+n\mathbb{Z}$ we run into the same problem, of all fields with $0$ characteristics multiplication by elements will take the elements out of the collection we have which it cannot do, it must be closed under scalar multiplication. The only possible choice that remains is our $\mathbb{F}_p$, but even there it doesn't matter because as mentioned before, it is not closed under summation and as such cannot form a vector space or even a module. The only possible exception is $0+n\mathbb{Z}$ however, I recommend you trying to find a scalar multiplication from $\mathbb{F}_p$ and see what happens, you'll find, if I am not mistake, that only the trivial multiplication leads to anything viable.

What we do have is that $n\mathbb{Z}$ does form a ring, group and a $\mathbb{Z}$-module, the other classes do not.

Answer 2

In mod 10 arithmetic positive integers having , say 3 as the unit digit, along with negative integers terminating in 7, form a residue class. What happens to the sum $13+23$?