Elements of generalized coordinate subgroups commute pairwise
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.3 Solution: Let $x = (x_j)$ and $y = (y_j)$. If $j \in I$, then $$(xy)_j =…
Exhibit all of the ring homomorphisms from the cartesian square of the integers to the integers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.5 Describe all ring homomorphisms from $\mathbb{Z} \times \mathbb{Z}$ to $\mathbb{Z}$. In each case, describe the kernel…
Generalized coordinate subgroups of a direct product
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.2 Solution: (1) Define a mapping $\varphi : G_I \rightarrow \times_{i \in I} G_i$ by $(\varphi(g))_i =…
The center of a direct product is the direct product of the centers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.1 Show that the center of a direct product is the direct product of the centers: $$Z(G_1…
Examine a given action of the additive real numbers on the Cartesian plane
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.3 Show that the additive group $\mathbb{R}$ acts on the $xy$-plane $\mathbb{R} \times \mathbb{R}$ by $r \cdot…
Direct product of rings is a ring under pointwise addition and multiplication
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.19 Let $I$ be a nonempty index set and let $R_i$ be a ring for each $i…
The diagonal subset of the Cartesian square of a ring is a subring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.18 Let $R$ be a ring. Prove that $\Delta(R) = \{(r,r) \ |\ r \in R \}$…
The Cartesian product of two rings is a ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.17 Let $R$ and $S$ be rings. Prove that the direct product $R \times S$ is a…
When the quotient by an intersection is isomorphic to a product of quotients
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.7 Let $G$ be a group and let $M,N \leq G$ be normal such that $G =…
Quotient of a product by a product is isomorphic to the product of quotients
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.4 Let $A$ and $B$ be groups, with $C \leq A$ and $D \leq B$ normal. Prove…