Abstract
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 LG-ring such that ¦ A M ¦ ≥ 3 for every maximal ideal M of A. Finally, these correspondences are homeomorphisms for convenient topologies. In the last section, we study the behaviour of bilinear and quadratic Witt rings by Galois extensions in the case where the base ring is a LG-ring. This question was quoted by M. Knebusch, A. Rosenberg and R. Ware for bilinear Witt rings (cf. [6], Proposition 5.11) in the semilocal case and by R. Baeza (cf. [1], Chapter V, Theorem 11.1) also in the semilocal case for both bilinear and quadratic Witt rings. But it seems to us that Baeza's proof presents a gap and we give here the solution in the case of LG-rings.
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