If you find any mistakes, please make a comment! Thank you.

## If a prime ideal contains a product of ideals then it contains one of the factors

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.11 Solution: Suppose $I \not\subseteq P$. Then there exists $a \in I$ such that $a \notin P$.…

## 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 unital ring, if the quotient corresponding to an ideal is a field, then the ideal is maximal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.5 Solution: Suppose we have a two-sided ideal $I$ with $M \subseteq I \subseteq R$. By the…

## A commutative unital ring is a field precisely when the zero ideal is maximal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.4 Solution: ($\Rightarrow$) Suppose $R$ is a field. Let $I \subseteq R$ be an ideal which properly…