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Consider the notion of finite representation type (FRT for short): An integral domain R has FRT if there are only finitely many isomorphism classes of indecomposable finitely generated torsion-free R-modules. Now specialize: Let R be of the form D+c O where D is a principal ideal domain whose residue fields are finite, c∈D is a nonzero nonunit, and O is the ring of integers of some finite separable field extension of the quotient field of D. If the D-rank of R is at least four then R does not have FRT. In this case we show that cancellation of finitely generated torsion-free R-modules is valid if and only if every unit of O /c O is liftable to a unit of O . We also give a complete analysis of cancellation for some rings of the form D+c O having FRT. We include some examples which illustrate the difficult cubic case.

Let F be a field of characteristic 0. We investigate the rationality of the center C n of the division algebra of two n × n generic matrices over F for n = 13 . If n is a prime power, then the only cases for which C n is known to be stably rational over F are n = 2 , 3 , 4 , 5 , and 7 with rationality proven for 2, 3, and 4. There is a certain Z S n -lattice, denoted A * , which has played an essential role in the proofs of these results. Formanek proved the case n = 4 by showing that C n is stably isomorphic to F ( A * − ) S n , the invariants of F ( A * − ) under the action of S n . Here A * − is A * ⊗ Z − and Z − is the sign representation of S n . Lebruyn and Bessenrodt proved the cases n = 5 and 7 by showing that C n is stably isomorphic to F ( A * ) S n . We show that for n = 13 there exists a Z S n -lattice M, which is stably permutation when restricted to the alternating group, such that F ( A * − ⊕ M ) S n and C n are stably isomorphic. We then show that a field extension of degree 2 of C n is stably isomorphic to a field extension of degree 2 of a rational extension of F.

For a F"q"^"2-maximal curve X of genus (q/n-1)q/2, where q/n is a nongap at some point, we show that X is F"q"^"2-isomorphic to the nonsingular model of a curve given by P(y)=A(x), where P is an additive separable polynomial of degree q/n and where A is a polynomial of degree q+1, under the further hypothesis that a certain field extension is Galois. In the particular case n=p=char(F"q"^"2) we were able to characterize such maximal curves without the assumption that a certain extension is Galois.

We define three-equipped posets, i.e. partially ordered sets equipped with three kinds of binary relations, and their representations and corepresentations over an inseparable cubic field extension F ⊂ G , in characteristic 3. These representations and corepresentations lead to some matrix problems of mixed type over the pair of fields ( F , G ). Through these problems of linear algebra, we completely describe in evident matrix form the indecomposables for some critical three-equipped posets, reducing the task for the rest of the critical sets to some matrix problems containing the pseudolinear pencil problem as a subproblem.

Let k0 be a field, chark0≠2, n⩾2, a,b 1 ,…,b n ∈k 0 ∗ such that the elements a , b 1 ,…, b n ∈k 0 ∗ /k 0 ∗ 2 are linearly independent. We show that there exists a field extension F/k0 and an anisotropic 4-dimensional form ϕ over F such that d±(ϕ)=a, the form ϕ F( b 1 ,…, b n ) is isotropic and the form (ϕ F( b 1 ,…, b n ) ) an is not defined over F.