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  Rings are among the fundamental structures studied in algebra. They are an intermediate stage between groups and fields, though they have a rich theory of their own. A ring, like a field, has two operations, addition and multiplication; the structure, again like a field, is a group under the operation of addition. The difference is in the operation of multiplication; in a ring, the elements do not have to have multiplicative inverses (another element which when multiplied by the original one gives the answer 1). This is because the integers form a ring but not a field, and much of the work in rings has been inspired by work in number theory. SMcL  



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