Geodesic
On Lattices in a Module Over a Matrix Algebra
Canadian mathematical bulletin, Tome 9 (1966) no. 1, pp. 57-61

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Let A be the matrix algebra of type n × n over a finite algebraic number field F, and V the module of matrices of type n × m over F. V is naturally an A-left module. Given a non-singular symmetric matrix S of type m × m over F, we have a bilinear mapping f of V on A such that f(x, y) = xSy' for elements x and y in V where y' is the transpose of y. In this case, corresponding to the arithmetic of A([l]), the arithmetical theory of V will be discussed to some extent as we establish the arithmetic of quadratic forms over algebraic number fields ([2]). In this note, we shall define a lattice in V with respect to a maximal order in A. and determine its structure (Theorem 1), and after giving a structure of a complement of a lattice (Theorem 2), we shall give a finiteness theorem of class numbers of lattices under some assumption (Theorem 3). 
Nobusawa, Nobuo. On Lattices in a Module Over a Matrix Algebra. Canadian mathematical bulletin, Tome 9 (1966) no. 1, pp. 57-61. doi: 10.4153/CMB-1966-007-4
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     title = {On {Lattices} in a {Module} {Over} a {Matrix} {Algebra}},
     journal = {Canadian mathematical bulletin},
     pages = {57--61},
     year = {1966},
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     number = {1},
     doi = {10.4153/CMB-1966-007-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-007-4/}
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