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## A ring has only the trivial one-sided ideals precisely when it is a division ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.6 Solution: ($\Rightarrow$) Suppose $R$ is a division ring. Let $L \subseteq R$ be a nonzero left…

## In a division ring, every centralizer is a division ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.10 Prove that if $D$ is a division ring, then $C_D(a)$ is a division ring for all…