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In the definition of a ring, it is nowhere stated that it must be closed under multiplication. But it seems to be true for all the examples of rings that I've seen so far. So, is this implicitly assumed in the definition or can it be proved?

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    $\begingroup$ What definition of ring are you using? $\endgroup$ – Michael Albanese Jul 15 '15 at 1:09
  • $\begingroup$ The one given in Dummit & Foote. That it must be an abelian group under addition, multiplication must be associative and the distributive law holds. $\endgroup$ – Train Heartnet Jul 15 '15 at 1:13
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    $\begingroup$ Think of it like this: In your universe, only the ring $R$ exists. So if you are allowed to multiply two things in $R$, and the result in NOT in $R$, then where could it be? You see that it doesn't even make sense. So intuitively, if you are allowed to multiply things, then the resulting element must be in your ring. $\endgroup$ – user46348 Jul 15 '15 at 1:18
  • $\begingroup$ This is explicitly stated in any proper definition of "ring". $\endgroup$ – hardmath Jul 15 '15 at 1:25
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A ring is an abelian group $R$ with an additional operation $\times$, that is, a function $\times:R\times R\to R$, satisfying the various axioms. The fact that this function has range $R$ is exactly the fact that $R$ is closed under multiplication.

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  • $\begingroup$ Oh! I completely missed the fact that $\times$ is a binary operation! Rather silly of me. Thank you! $\endgroup$ – Train Heartnet Jul 15 '15 at 1:16
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I suspect you are misreading the definition. The definition I know says that multiplication is a "binary operation" on the underlying set which means that it "$ab$" is a mapping from $X \times X\rightarrow X$. That is, the result of "$ab$", for $a$ and $b$ two members of the ring is again a member of the ring. I.e. it is closed under multiplication.

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Yes, by the definition of a ring.

A ring $(R,+,\cdot)$ satisfies eight properties: five of which is that $(R,+)$ is an abelian group (closed, associative, existence of $0$, invertibility of each element, and commutativity), two of which is that $(R,\cdot)$ is a semi-group (closed and associative) and the eighth property is the distributive property.

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