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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
2
votes
Finding purely transcendental parts of field extensions
I don't know if this is a counterexample or not, but what would you do if $K$ is the field of fractions of the monoid ring $R = \mathbb{Z}[\prod_{i=1}^\infty \mathbb{N}]$? ($R$ is an integral domain b …
10
votes
Integrally closed
I claim the integral closure of the ring $R:=F+xk[[x]]$ is $R':=F'+xk[[x]]$, where $F'$ is the algebraic closure of $F$ inside $k$. For one, the field of fractions of $R$ is $k((x))$, and $F'$ is inte …