Do I Have to Show All Axioms to Prove a Set is a Ring?

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In summary, when showing that a set S is a ring, all axioms must be demonstrated, even if some may seem trivial. However, when showing that S is a subring, only the requirements for a subring need to be shown, as it already satisfies all the ring axioms as a subset of a ring.
  • #1
evinda
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Hey again! :)
I have a question..
If I have to show that a set $S$ is a ring,do I have to show all the axioms or is it enough to show the criteria:
$s_1,s_2 \in S$ and
  • $s_1-s_2 \in S$
  • $s_1 \cdot s_2 \in S$
?
 
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  • #2
evinda said:
Hey again! :)
I have a question..
If I have to show that a set $S$ is a ring,do I have to show all the axioms or is it enough to show the criteria:
$s_1,s_2 \in S$ and
  • $s_1-s_2 \in S$
  • $s_1 \cdot s_2 \in S$
?

Welcome back! ;)

I'm afraid you have to show all the axioms.
Luckily most axioms are often enough fairly trivial.
Still, you have to show all of them, and every now and then one of them holds a surprise, turning everything upside down.

It's a different matter if you have to show something is a subring.
Then you only have to show what is required for a subring.
Effectively you are borrowing from the fact that it is a subset of a ring for which all the ring axioms are already satisfied.
 
  • #3
I like Serena said:
Welcome back! ;)

I'm afraid you have to show all the axioms.
Luckily most axioms are often enough fairly trivial.
Still, you have to show all of them, and every now and then one of them holds a surprise, turning everything upside down.

It's a different matter if you have to show something is a subring.
Then you only have to show what is required for a subring.
Effectively you are borrowing from the fact that it is a subset of a ring for which all the ring axioms are already satisfied.

I understand...Thanks! :)
 

FAQ: Do I Have to Show All Axioms to Prove a Set is a Ring?

What is a ring in mathematics?

A ring is a mathematical structure that consists of a set of elements and two binary operations (usually addition and multiplication). It must satisfy certain properties, such as closure, associativity, and distributivity, in order to be considered a ring.

How do you prove that a set is a ring?

In order to prove that a set is a ring, you must show that it satisfies all the properties of a ring. This includes closure under addition and multiplication, associativity of both operations, the existence of an identity element for addition, and the existence of inverse elements for addition. You must also show that the set is commutative under addition and has distributivity between addition and multiplication.

What are the basic properties of a ring?

The basic properties of a ring include closure, associativity, commutativity, distributivity, and the existence of identity and inverse elements. These properties ensure that the set and operations within the ring behave in a consistent and predictable manner.

Can a ring have more than two operations?

Yes, a ring can have more than two operations. In addition to addition and multiplication, a ring can also have other operations such as exponentiation or composition. However, it must still satisfy all the properties of a ring in order to be considered as such.

How is a ring different from a field?

A ring and a field are both algebraic structures, but a field has additional properties that a ring does not. A field must have a multiplicative inverse for every element (except for the additive identity), while a ring only requires a multiplicative inverse for a subset of its elements. Additionally, a field must also be commutative under multiplication, while a ring does not necessarily have to be.

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