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I'm studying Abstract Algebra right now, currently covering rings. In the introduction of the subject, I am curious as to why there is no need for there to be a multiplicative identity. I understand that in order to be a ring, we require the set to be an abelian group under addition operation and a monoid under multiplication. But what is the reason for the monoid, rather than group under multiplication--or lack of multiplication?

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    You seem to be asking two questions - are you asking about the multiplicative identity or multiplicative inverses? For the former, see http://math.stackexchange.com/questions/48587/definition-of-ring-vs-rng . For the latter, well, that's what fields are for.2011-07-24
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    Related questions [here](http://math.stackexchange.com/questions/16168/applications-of-rings-without-identity) and [here](http://math.stackexchange.com/questions/48587/definition-of-ring-vs-rng).2011-07-24
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    It's the former. Thanks both for the links.2011-07-24
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    Also: http://mathoverflow.net/questions/22579/2011-07-24
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    See also this question on [examples/motivation for non-unital rings.](http://math.stackexchange.com/questions/37705/non-unital-rings-a-few-examples/37716#37716) Note that the definition of a ring also requires that the additive and multiplicative structures are related - by the distributive law. Without such one would simply have a set with two completely unrelated structures.2011-07-25
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    Regarding the statement of the question: if the ring has a multiplicative identity then it is a monoid under multiplication. If a multiplicative identity does not exist then the ring (also called general ring) is not a monoid under multiplication.2011-07-25

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