Determine whether or not a subset is an ideal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.8 Solution: (1) Note that $(1,1) \in D$. However, $(1,0)(1,1) = (1,0) \notin D$. Since $D$ does…
Prove that a given function is a ring homomorphism and describe its kernel
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.7 Solution: Let $A,B \in R$ be arbitrary, with $A = \begin{bmatrix} a_1 & b_1 \\ 0…
Determine whether or not a mapping is a ring homomorphism
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.6 Decide which of the following are ring homomorphisms from $M_2(\mathbb{Z}) to \mathbb{Z}$. (1) $\begin{bmatrix} a &…
Exhibit all of the ring homomorphisms from the cartesian square of the integers to the integers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.5 Describe all ring homomorphisms from $\mathbb{Z} \times \mathbb{Z}$ to $\mathbb{Z}$. In each case, describe the kernel…
Computation of all ring homomorphisms from Z to Z/(30)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.4 Find all ring homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}/(30)$. In each case describe the kernel and the…
Find all the ring homomorphic images of the integers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.3 Find all homomorphic images of $\mathbb{Z}$. Solution: Recall that every additive subgroup of $\mathbb{Z}$ is cyclic,…
Z[x] and Q[x] are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.2 Prove that the rings $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic. Proof: In $\mathbb{Q}[x]$, $f(x)+f(x)=g(x)$ has a…
2Z and 3Z are not isomorphic as rings
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.1 Prove that the rings $2\mathbb{Z}$ and $3\mathbb{Z}$ are not isomorphic. Solution: Suppose $\varphi : 2\mathbb{Z} \rightarrow…
Generalized coordinate subgroups of a direct product
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.2 Solution: (1) Define a mapping $\varphi : G_I \rightarrow \times_{i \in I} G_i$ by $(\varphi(g))_i =…
The center of a direct product is the direct product of the centers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.1 Show that the center of a direct product is the direct product of the centers: $$Z(G_1…