Basic properties of quotients of a polynomial ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.14 Solution: We begin with a lemma. Lemma: Let $R$ be a commutative ring with $1 \neq…
The complete homomorphic preimage of a prime ideal is a prime ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.13 Solution: (1) By Exercise 7.3.24, $\varphi^\ast[P]$ is an ideal of $R$. Now suppose $ab \in \varphi^\ast[P]$.…
The product of two finitely generated ideals is finitely generated
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.12 Solution: ($\subseteq$) Let $x = \sum_i r_is_i \in IJ$, where $$r_i = \sum_j t_{i,j}a_j$$ and $$s_i…
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 $a \notin P$.…
If a nontrivial prime ideal contains no zero divisors, then the ring is an integral domain
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.10 Solution: Let $a,b \in R$ such that $ab = 0$. Since $P$ is a prime ideal…
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…