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

[evinda]

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

• $s_1-s_2 \in S$
• $s_1 \cdot s_2 \in S$

?

 

[I like Serena]

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.

 

[evinda]

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! :)