Algebraic Number Theory
The leading example is the case where A is a number field K and 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 field extension A=Q(i) of Gaussian rationals over Q, the integral closure of Z is the ring of Gaussian integers Z and so this is the unique maximal Z-order: all other orders in A are contained in it: for example, we can take the subring of the
for which b is an even number.
The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.
Read more about this topic: Order (ring Theory)
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