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Basic properties of n-ary direct products of groups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.16 Solution: (1) First we show that $\iota_k$ is an injective homomorphism. If $g,h \in G_k$, then…

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 =…

The powerset of a set is a Boolean ring under intersection and symmetric difference

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.21 Let $X$ be a nonempty set and let $R = \mathcal{P}(X)$ denote the power set of…

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…

Every Boolean ring is commutative

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.15 A ring $R$ is called Boolean if $a^2 = a$ for all $a \in R$. Prove…