Proving Properties of an Ordered Ring: R+ & R

In summary, an ordered ring is a mathematical structure with elements, operations, and an ordering relation. It differs from a regular ring by having an ordering relation, and must satisfy properties such as closure, commutativity, and existence of identity and inverses. Proving these properties is important to validate the structure and ensure reliable mathematical reasoning.
  • #1
invisible_man
16
0
Let R be an ordered Ring. Assume R+ is well-ordered
Prove:
a) min(R+) = 1.
b) R is an integer ring
 
Physics news on Phys.org
  • #2
Sounds like a homework problem all right.

You didn't say it, but I assume you're looking for help? What have you done (successful or not), and where are you stuck?
 
  • #3
(1) Why does min(R+) exist?
(2) Let u = min(R+), assume it is not 1, try to get a contradiction.
 

1. What is an ordered ring?

An ordered ring is a mathematical structure consisting of a set of elements, operations such as addition and multiplication, and an ordering relation. The ordering relation allows for the comparison of elements in the ring, such as determining which is larger or smaller.

2. What are the properties of an ordered ring?

An ordered ring must satisfy the following properties: closure under addition and multiplication, commutativity and associativity of addition and multiplication, existence of an identity element for addition and multiplication, existence of additive and multiplicative inverses, distributivity of multiplication over addition, and the ordering relation must be transitive, reflexive, and compatible with addition and multiplication.

3. How is an ordered ring different from a regular ring?

The key difference between an ordered ring and a regular ring is the presence of an ordering relation. In a regular ring, elements cannot be compared in terms of size or magnitude, whereas in an ordered ring, the ordering relation allows for such comparisons.

4. How do you prove the properties of an ordered ring?

To prove the properties of an ordered ring, we must use a combination of algebraic manipulations and logical reasoning. This involves showing that each property holds for all elements in the ring, using the definition of an ordered ring and the properties of addition and multiplication.

5. Why is it important to prove the properties of an ordered ring?

Proving the properties of an ordered ring is important because it allows us to establish the necessary conditions for a structure to be considered an ordered ring. This helps to ensure that our mathematical reasoning and calculations are valid and reliable when working within the context of ordered rings.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
769
  • Linear and Abstract Algebra
Replies
11
Views
1K
  • Linear and Abstract Algebra
2
Replies
55
Views
3K
  • Linear and Abstract Algebra
Replies
2
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
936
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
17
Views
4K
Replies
6
Views
1K
Back
Top