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Let L be a finite dimensional,
symmetric algebra with 1 over the quotient field K of an infinite domain R. When L
is quaternion algebra and R is a Prüfer ring, it is known that an R-module
contained in L is invertible if and only if it satisfies a certain relation between it, its
dual and its discriminant [7]. We call a symmetric algebra L with this property a
Brandt algebra.
We prove: (i) If R is a Prüfer ring, then L is a Brandt algebra if it is
3-dimensional or if it contains only invertible modules. (ii) If R is a valuation ring
and L is a Brandt algebra, then any module or its dual is invertible. (iii) If
R≠K, then L is not a Brandt algebra if it is generated over K by a single
non-cubic element or if it is a matrix algebra over a symmetric algebra L1 with
(L1 : K) > 1.
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