High Quality Content by WIKIPEDIA articles! In mathematics, an order in the sense of ring theory is a subring mathcal{O} of a ring R that satisfies the conditions.The third condition can be stated more accurately, in terms of the extension of scalars of R to the real numbers, embedding R in a real vector space (equivalently, taking the tensor product over mathbb{Q}). In less formal terms, additively mathcal{O} should be a free abelian group generated by a basis for R over mathbb{Q}. The leading example is the case where R is a number field K and mathcal{O} is its ring of integers. In algebraic number theory there are examples for any K other than the rational field of proper subrings of the ring of integers that are also orders. For example in the Gaussian integers we can take the subring of the a + bi, for which b is an even number. A basic result on orders states that the ring of integers in K is the unique maximal order: all other orders in K are contained in it.

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