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We deal with the existence of self-dual normal basis for Galois extensions of a commutative ring. We consider commutative rings which are local, connected semi-local (under some suitable restrictions) or zero-dimensional. We show that for such kind of rings every Galois extension of odd degree has a self-dual normal basis.
It is well known that if A is a semilocal ring, there exists a one to one correspondence between the set of signatures of A and the set of minimal prime ideals of the bilinear Witt ring W(A). We show that this correspondence also holds if A is an LG-ring. Moreover, there exists a one to one correspondence between the set of minimal prime ideals of W(A) and the set of maximal orders of A, if A is an...
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