What does mean?
Z[X] must be the ring of all polynomials with coefficients in Z.
This makes an ideal ???
And then is quotient ring?
Is it possible this notation to denote something other ?
An ideal is a set.
The denominator of a quotient ring also has to be a set.
is not a set, but an element of Z[X].
So I'd interpret it as a set of polynomials divided by :
If it is supposed to mean anything else, it is bad notation IMO and deserves a big red cross through it.
This is from Wikipedia article about quotient rings.
Quotient ring - Wikipedia, the free encyclopedia
I'm reading and I'm reading and I'm reading and still cannot imagine how it all fits together. For example it seems that in the ring which seems to make sense in the context of the above citation. We "make" a ring which satisfies this property. But I cannot still imagine the cosets and their representatives...Now consider the ring R[X] of polynomials in the variable X with real coefficients, and the ideal consisting of all multiples of the polynomial . The quotient ring is naturally isomorphic to the field of complex numbers C, with the class [X] playing the role of the imaginary unit i. The reason: we "forced" , i.e. , which is the defining property of i.
I see.
Well, a coset would be a polynomial in Z[X] plus a multiple of .
A representative is any polynomial, with perhaps a multiple of subtracted, so there are no powers of 4 anymore (for easier comparison).
.
the notation (f(x)) is often used for the principal ideal generated by f(x), that is: all polynomials of the form k(x)f(x), for k(x) in Z[x].
note that the cosets p(x) + (f(x)) consist of the cosets of those polynomials p(x) where deg(p(x)) < deg(f(x)) (since we can use the division algorithm to write
p(x) = q(x)f(x) + r(x) in which case p(x) + (f(x)) = [r(x) + (f(x))] + [q(x)f(x) + (f(x))] = r(x) + (f(x)) since q(x)f(x) is in (f(x)) so that q(x)f(x) + (f(x)) = 0 + (f(x)) = (f(x)) itself).
the "classic example" is where f(x) = x^{2} + 1. note in this case, if we write I = (x^{2} + 1), in Z[x]/I we have cosets of the form:
a + bx + I, where a,b are integers. also note that for the coset x + I:
(x + I)(x + I) + (1 + I) = (x^{2} + I) + (1 + I) = (x^{2} + 1) + I = I, the 0-element of Z[x]/I.
thus a + bx + I ↔ a + bi is a (ring) isomorphism of Z[x]/I with Z[i], the gaussian integers (replacing "polynomials in x" with "polynomials in i" (i being a square root of -1)).