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The homework question is "Is $S = \{(a,b) \mid a + b = 0\}$ a subring of $\mathbb Z \times \mathbb Z$? Justify your answer"

Do I do this by checking the 8 axioms of a ring? If so how is the $\mathbb Z\times \mathbb Z$ part put into play here?

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You can either check all the axioms, or else the abbreviated set of axioms that checks if $S$ is a subring of $R$. If you'd like to learn why the following works, I suggest you begin with this solution by Pete Clark which will lead you to a wiki article discussing the topic.

  • $S$ is nonempty
  • For all $a,b\in S$, $a-b\in S$
  • For all $a,b\in S$, $ab\in S$
  • (If your definition of subring requires that the set contain the identity of the big ring) the identity of $R$ is in $S$

Checking these the first three is sufficient if your definition of subring doesn't require sharing identity, and checking the last three is sufficient if you do require that.

You ought to quickly discover that the first two points are satisfied by $S$, but the last two points are not.

$\Bbb Z\times \Bbb Z$ comes into play because it gives you a concrete operations that you are familiar with to check. It's just regular integer addition and multiplication coordinatewise.

Actually, you'll get the same answer if you use $R\times R$ for any ring $R$, just so long as $2\neq 0$ in that ring. Here, $R=\Bbb Z$ is a special case. If $2=0$, you'll actually find the subset is a subring.

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