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Let $k$ be a field. In 1941, Jung showed that all polynomial $k$-algebra automorphisms of the rational (polynomial) functions in two variables, denoted by $k(x,y)$ can be written as compositions of th …

Let $R$ be a ring of (smooth?) functions on a (connected?) manifold of dimension $n$. What concept of dimension (of the ring $R$) gives the dimension of the manifold? To what class of rings does this …

Let $R$ be a commutative ring and let $M,N$ be two finitely generated projective $R$-modules which have equal rank (not necessarily constant). What kind of general results are there concerning the que …

Let $A$ be a commutative algebra (over the complex numbers, with a unit) and let $M$ be a finitely generated projective $A$-module, and let $m_1,\ldots,m_n$ be a set of generators of $M$. The Dual Bas …

Assume that $R$ is a commutative ring with a ring compatible ordering and let $A$ and $B$ be symmetric $n\times n$ matrices with entries in $R$ such that $\sum x_iA_{ij}x_j\geq 0$ and $\sum x_iB_{ij}x …