Not all ideals are prime
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.9 Solution: First we show that $I$ is an ideal. To That end, let $f,g \in I$.…
In an integral domain, two principal ideals are equal precisely when their generators are associates
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.8 Solution: ($\Rightarrow$) Suppose $(a) = (b)$. Then $a \in (b)$, and we have $a = ub$…
In a polynomial ring, the ideal generated by the indeterminate is prime precisely when the coefficient ring is an integral domain
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.7 Solution: We begin with a lemma. Lemma: Let $R$ be a ring. Then $\varphi : R[x]…
A ring has only the trivial one-sided ideals precisely when it is a division ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.6 Solution: ($\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…
Find a generating set for the augmentation ideal of a group ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.2 Solution: Recall that the augmentation ideal of $R[G]$ is the kernel of the ring homomorphism $R[G]…
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…
Exhibit two Sylow 2-subgroups of Symmetric group $S_5$
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.8 Solution: $|S_5| = 120 = 2^3 \cdot 3 \cdot 5$, so that the Sylow 2-subgroups of…
Exhibit all of the Sylow 2-subgroups of Symmetric group $S_4$
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.7 Solution: The Sylow 2-subgroups of $S_4$ have order 8, and by Sylow’s Theorem, $n_2 = 1…