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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…

## Characterize the ideals consisting of all matrices with a single nonzero column

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.1 Solution: In Exercise 7.2.6, we saw that $AE_{i,j}$ is the matrix whose j-th column is the…

## In a ring, sets of left and right annihilators are one-sided ideals

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.22 Solution: We begin with a definition and some lemmas. Definition: Let $R$ be a ring, \$A…