What does integer-like mean?

[Haydon Berrow]

I had a conversation with someone who claimed that a certain ring was integer-like; exactly who is immaterial. He was unable to express what he meant by this but it did make me wonder where the name of the ring of algebraic integers comes from.

My guess is that we have the field of algebraic numbers and they have the property that any algebraic number can be expressed as the ratio of two algebraic integers* exactly as any rational number can be expressed as the ratio of two integers. The analogy is very good.

*to be more precise, if z is an algebraic number then there exists algebraic integers x and y such that x = y*z. In fact this can be strengthened and x and y can be found such that y is an integer, not an algebraic integer.

Given this, a reasonable interpretation of "R is an integer-like ring" is that R is the ring of integers for some field F in the sense that F is the field of rationals for the ring R. It's still a bit unsatisfactory though because I haven't constructed the ring of algebraic integers from the field of algebraic numbers, the definition of the ring comes from elsewhere and it is shown that it has this property. Is there a process for constructing the "ring of integers in a field" and what properties does such a field have to have for it to exist? Are there other famous examples?

[Dan Christensen]

For a straightforward question like this (I don't have the answer), I can recommend Math Stackexchange. It is a more tightly moderated forum with a fairly rigid Q & A format. Vague or open-ended questions and extended discussions are discouraged.

Dan

(Moderator)

[Dan Christensen]

You may find the following Peano-like axioms for the set of integers to be helpful:

1.  0 in Z

2.  S is a bijection on Z

3.  NN is a subset of Z  (non-negative integers)

4.  NP is a subset of Z  (non-positive integers)

5.  Z is the union of NN and NP

6.  {0} is the intersection of NN and NP

7.  S is closed on NN

8.  For all x in NN: S(x) =/= 0

9.  S^-1 is closed on NP

10.  For all x in NP: S^-1(x) =/= 0

11.  For all subsets P of Z: [0 in P  &  For all x in P :[[x in NN => S(x) in P] & [x in NP => S^-1(x) in P]] => P=Z]   (2-way induction)

Any structure (Z, NN, NP, S, 0) that satisfies the above axioms might be said to be "integer-like." 

Dan

[Dan Christensen]

On Sunday, 3 February 2019 14:55:48 UTC-5, Dan Christensen wrote:

You may find the following Peano-like axioms for the set of integers to be helpful:

1.  0 in Z

2.  S is a bijection on Z

3.  NN is a subset of Z  (non-negative integers)

4.  NP is a subset of Z  (non-positive integers)

5.  Z is the union of NN and NP

6.  {0} is the intersection of NN and NP

7.  S is closed on NN

8.  For all x in NN: S(x) =/= 0

9.  S^-1 is closed on NP

10.  For all x in NP: S^-1(x) =/= 0

11.  For all subsets P of Z: [0 in P  &  For all x in P :[[x in NN => S(x) in P] & [x in NP => S^-1(x) in P]] => P=Z]   (2-way induction)

11.  (In other words)  Every element of NN can be reached by going from one number to its successor starting at 0 AND every element of NP can be reached by going from one number to its predecessor starting at 0.

Any structure (Z, NN, NP, S, 0) that satisfies the above axioms might be said to be "integer-like." 

Comments from other readers?

Dan