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## 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 noncommutative ring, the set of nilpotent elements need not be an ideal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.31 Solution: We begin with a lemma. Lemma: Let $R$ be a ring with $1 \neq 0$.…

## Characterize the two-sided ideals of a matrix ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.21 Solution: First, let $I \subseteq M_n(R)$ be a two-sided ideal. Let $J \subseteq R$ consist of…

## Embed a ring of quadratic integers in a ring of matrices

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.12 Solution: We begin with a lemma. Lemma: If $D \in \mathbb{Z}$ is not a perfect square,…