Quotient ring of poly ring Z[x]

[jakelyon]

Can anyone explain, in detail, why/why not Z[X]/(2x) is isomorphic to Z/2Z? I know that every element in Z[x] can be written as a_0 + a_1 x + a_2 x^2 + ... with a_i in Z and only finitely many a_i's are nonzero. Now, does (2x) = (2, 2x, 2x^2,...)? Also, the quotient is "like" taking 2x=0, or x=0. Thus, I think that all elements of Z[x]/(2x) would look like a_0/2 for some a_0 in Z. But this does not give Z/2Z does it? Thanks.

 

[daveyinaz]

What is the cardinality of Z/2Z? Then look at the size of Z[x]/(2x).

 

[jakelyon]

Thanks for replying daveyinaz, but I am not sure I am following. However, I been doing some reading:

(2x) is the ideal consisting of all linear combinations of 2x (with integer coefficients). Now, by moding Z[x] out by (2x) it is "like" sending x to 0. So, if I am correct, then Z[x]/(2x) = Z[0] =
Z, not Z/2Z, right?

Does this make sense? How would I get Z/2Z then? Thanks.

 

[eok20]

You are right that you can think of the quotient as forcing 2x = 0. However, this does not mean that x = 0, it means that all multiples of 2x by elements in Z[x] are equal to zero. So x is not zero since x is not a multiple of 2x in Z[x]. Also, all of Z will be in that quotient since no integer is a multiple of 2x in Z[x]. Things in Z[x]/(2x) will look like polynomials with any constant term but with coefficients on the other terms being 0 or 1.

Maybe part of your trouble is thinking that all rings are integral domains (integral domains are rings such that if ab = 0 then a = 0 or b = 0). This ring is not and there are more familiar ones that are not either such as the ring of all matrices.

If you want to get Z/2Z from a quotient of Z[x] you would have to quotient out by the ideal (2,x). Note that Z[x] is not a principal ideal domain and this ideal cannot be generated by a single element.

 

[blue_raver22]

Firstly, let's define some terms for clarity:

- Poly ring Z[x]: This refers to the set of all polynomials with coefficients in the integers (Z) and variable x.
- Quotient ring: This is a mathematical structure formed by taking a ring (a set with operations of addition and multiplication) and a subset of that ring, and creating a new structure where the elements of the subset are considered to be equivalent to each other. This is denoted by using the notation of a ring followed by a slash and the subset in parentheses.
- Z/2Z: This is the ring of integers modulo 2, also known as the integers under addition and multiplication modulo 2.

Now, let's consider the given question of whether Z[x]/(2x) is isomorphic to Z/2Z. Isomorphism is a concept in mathematics that describes a relationship between two structures where they have the same underlying structure, but their elements may be different. In this case, we are interested in whether the two structures have the same algebraic properties.

To determine if two structures are isomorphic, we need to find a function (called an isomorphism) that maps elements from one structure to the other in a way that preserves the algebraic properties. In this case, we are looking for a function that maps elements from Z[x]/(2x) to Z/2Z. Let's call this function f.

Now, let's consider the elements of Z[x]/(2x). As you correctly stated, every element in Z[x] can be written as a_0 + a_1 x + a_2 x^2 + ... with a_i in Z and only finitely many a_i's are nonzero. However, when we take the quotient by (2x), we are essentially setting 2x equal to 0. This means that in Z[x]/(2x), any polynomial that has a term with x^2 or a higher power of x can be simplified to 0. So, in this quotient ring, the elements only have terms up to a_1 x. This can be written as a_1 x + a_0, where a_0 and a_1 are integers. So, we can say that the elements of Z[x]/(2x) can be written as a_1 x + a_0, where a_0, a_1 are integers.

Now,