Puppy Linux Discussion Forum

[mahaju]

This is an excerpt from an article I am reading. In the 2nd line the Z refers to set of integers I think, but what does Z/m mean? Does it mean the set {0,1, ..., m-1} (ie, the set of integers mod m) or does it mean something else?
Please help me out here.
Thanks in advance.



Also, a homomorphish which is one to one and onto (bijective) is called isomorphism. Is this true?

[myke]

In mathematics, Z/m indeed refers to the set {0,1, 2, ..., m-1} but generally mathematicians are more interested in looking at abstract groups, rings and fields. The group {Z/m,+} refers to the set Z/m with addition modulo m. So, the subgroup {Z/4,+} would be {0,1,2,3} with 3+1=0, 1+2=3, 2+3=1, etc. You can think of it as adding the remainders of two numbers so the remainder of 11 after dividing by 4 + the remainder of 9 after dividing by 4 = the remainder of 20 after dividing by 4. It forms a commutative group with the identity being 0. The meaning I gave it is in some sense canonical. (The technical term is isomorphism).

If you look at {Z/m,+, ⊗} , with the addition and multiplication being modulo m, you get a commutative field with the additive identity being 0 and the multiplicative identity being 1. In other European languages, field is generally not used in this context, French: corps, German: Körper, Dutch: lichaam, but Flemish: veld.

[mahaju]

Thank you very much
Considering the symbol to mean the set Z mod m the other parts of the article make a little more sense, but I still have many problems with this. For instance, do you have any idea what it means in line 5, that
Steps 1 and 3 are "no-ops" if Z/m is represented as {0,1,...,m-1}
What is "no-ops" and why the "if"? ( since Z/m means {0,1,...,m-1} )

Again, in this next section, R[x] means the set of all polynomials in R with variable x, that is, the coefficients of the variables in the polynomials are members of the ring R. So does that mean R[y][x] (line 2) is the set of polynomials of two variables x and y? And what is the difference between writing R[y][x] and R[x][y], because in later sections the articles hunts that they are not the same, however it's not clear how they are different. Also, by analogy, R[y][x]/(x^n-y) would mean a set of polynomials that we get as remainder when we divide the members of the set in the numerator by the denominator. Is this right? Do I have to visualize this new set as something being finite like Z/m, or is it possible for this new set to have infinite members?

Thanks