Vol. 32, No. 2, 1970

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The Brandt condition and invertibility of modules

Howard Gorman

Vol. 32 (1970), No. 2, 351–371

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 RK, 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.

Mathematical Subject Classification
Primary: 16.40
Received: 26 September 1968
Published: 1 February 1970
Howard Gorman