We prove that deformations of tame Krull-Schmidt bimodule problems with trivial differential are again tame. Here we understand deformations via the structure constants of the projective realizations which may be considered as elements of a suitable variety. We also present some applications to the representation theory of vector space categories which are a special case of the above bimodule problems.

In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in M_n\left(I\right)$, there exist left invertible $U_1,U_2\in M_n\left(R\right)$ and right invertible $V_1,V_2\in M_n\left(R\right)$ such that $U_1{V}_{1}A{U}_2{V}_2=diag(e_1,\cdots ,e_n)$ for idempotents $e_1,\cdots ,e_n\in I$.

Our article contributes to the classification of dissident maps on ℝ ⁷, which in turn contributes to the classification of 8-dimensional real division algebras. We study two large classes of dissident maps on ℝ ⁷. The first class is formed by all composed dissident maps, obtained from a vector product on ℝ ⁷ by composition with a definite endomorphism. The second class is formed by all doubled dissident maps, obtained as the purely imaginary parts of the structures of those 8-dimensional...