Subring Math Problem: Find Number of Elements in \mathbb{Z}_{2000} Subring

  • Thread starter roam
  • Start date
In summary: Could you please explain a bit more and maybe give some examples? Because I'm very confused... I think all of the elements which will be zero must be factors of 2000=2.52.17. I mean 850 x (something x k)=2000k' Do I need to try these:Elements of S which are zero are factors of 2000=2.52.17. So, for instance, 400x24.5x=4000k. 200x23.52x=2000k, and 1000x24.53x=4000k.
  • #1
roam
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Homework Statement



Let R be a ring and a be an element of R. Let [tex]S= \left\{ x \in R: ax=0_R \right\}[/tex]. S is a subring of R.

Let [tex]R= \mathbb{Z}_{2000}[/tex] and [tex]a=850[/tex]. Determine the elements of the subring S as defined previously. How many elements are in S?


The Attempt at a Solution



The elements of the subring S will be elements x from [tex]\mathbb{Z}_{2000}[/tex] such that [tex]850.x=0_R[/tex].

And I think since 850x=0-5000n, [tex]x= \frac{2000}{850} n = \frac{40}{21} n[/tex] then

n=k.21

But what I do I need to do to find the number of elements in S? Is there a quick way of finding this?
 
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  • #2


What's the largest multiple of 21 in Z/2000? What k does this correspond to? The other elements of S follow from this.
 
  • #3


fzero said:
What's the largest multiple of 21 in Z/2000? What k does this correspond to? The other elements of S follow from this.

The largest multiple of 21 [tex]\mathbb{Z}_{2000}[/tex] is 1995. It ocrresponds to k=95, since 21x95=1995. What do you mean "other elements of S follow from this"? How do I need to figure out how many elements are in S?
 
  • #4


roam said:
The largest multiple of 21 [tex]\mathbb{Z}_{2000}[/tex] is 1995. It ocrresponds to k=95, since 21x95=1995. What do you mean "other elements of S follow from this"? How do I need to figure out how many elements are in S?

Sorry, I thought your method actually determined the elements of S. I would look at the prime factorizations 2000 = 24 53, 850 = 2 52 17. Now, by comparing the prime factorizations, what is the smallest x (call it xg) such that a xg = 0R? Now note that all multiples m xg are also in S.
 
  • #5


fzero said:
Sorry, I thought your method actually determined the elements of S. I would look at the prime factorizations 2000 = 24 53, 850 = 2 52 17. Now, by comparing the prime factorizations, what is the smallest x (call it xg) such that a xg = 0R? Now note that all multiples m xg are also in S.

Firstly, how do you determine this xg from the prime factorization? Also, how does it help to determine the number of elements in S?
 
  • #6


roam said:
Firstly, how do you determine this xg from the prime factorization? Also, how does it help to determine the number of elements in S?

The condition [tex]a x =0_R[/tex] requires that [tex]a x = 2^4 5^3 k[/tex] for some k. Looking at the prime factors in a leads us to conclude that [tex]x \in S[/tex] have the form [tex]x_{k'}=2^m 5^n k'[/tex]. [tex]m,n[/tex] are easily determined, while the [tex]k'=1,\ldots k'_{\text{max}}[/tex] are constrained by the condition that [tex]x\in R[/tex].
 
  • #7


fzero said:
The condition [tex]a x =0_R[/tex] requires that [tex]a x = 2^4 5^3 k[/tex] for some k. Looking at the prime factors in a leads us to conclude that [tex]x \in S[/tex] have the form [tex]x_{k'}=2^m 5^n k'[/tex]. [tex]m,n[/tex] are easily determined, while the [tex]k'=1,\ldots k'_{\text{max}}[/tex] are constrained by the condition that [tex]x\in R[/tex].

Could you please explain a bit more and maybe give some examples? Because I'm very confused... I think all of the elements which will be zero must be factors of 2000=2.52.17. I mean 850 x (something x k)=2000k' Do I need to try these:

400x 24.5 x k
200 x 23.52 k
1000 x 24.53 k

for different k's.
 
  • #8


Since [tex]850 = (2) (5^2) ( 17)[/tex], we compute

[tex] a x_{k'} = 2^{m+1} 5^{n+2} 17 k' .[/tex]

This is [tex]0 (\mod 2000)[/tex] if [tex]m=3,n=1[/tex], so

[tex]x_{k'} = 40 k'.[/tex]

Note that (850)(40)=(17)(2000), so your intuition is correct. Since 17 is prime, 40 is the smallest integer for which this works. Now [tex]x_{50} = 2000 = 0_R[/tex] gives us [tex]k'_{\text{max}}[/tex].
 

1. What is a subring in mathematics?

A subring is a subset of a given ring that is closed under addition, subtraction, and multiplication. This means that the elements of a subring, when operated on with the same operations as the original ring, will result in an element within the subring.

2. How do you find the number of elements in a subring of \mathbb{Z}_{2000}?

To find the number of elements in a subring of \mathbb{Z}_{2000}, you need to determine the elements that satisfy the criteria of being closed under addition, subtraction, and multiplication. This can be done by identifying the common factors of 2000 and selecting the elements that have a common factor with 2000. For example, if the subring contains elements that are multiples of 5, then there are 2000/5 = 400 elements in the subring.

3. What is the significance of \mathbb{Z}_{2000} in this problem?

\mathbb{Z}_{2000} is the ring of integers modulo 2000, which means that it contains all integers from 0 to 1999. It is significant in this problem because the elements of the subring must be selected from this set of integers.

4. How do you know if a subring is a proper subset of the original ring?

A subring is a proper subset of the original ring if it does not contain the identity element of the original ring or if it does not contain all elements of the original ring. In other words, if there are elements in the original ring that do not satisfy the criteria for being in the subring, then the subring is a proper subset.

5. Can a subring have an infinite number of elements?

Yes, a subring can have an infinite number of elements. For example, the set of all multiples of 2 within \mathbb{Z}_{2000} is a subring with an infinite number of elements.

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