Chapter 3.1, Problem 52E

Let $G_1$ and $G_2$ be groups with respect to addition. Define equality and addition in the Cartesian product by $G_1\times G_2$

$(a,b)=(a^{\prime },b^{\prime })$ if and only if $a=a^{\prime }$ and $b=a^{\prime }$

$(a,b)+(c,d)=(a\oplus c,b⊞d)$

Where $\oplus$ indicates the addition in $G_1$ and $⊞$ indicates the addition in $G_2$.

Prove that $G_1\times G_2$ is a group with respect to addition.

Prove that $G_1\times G_2$ is abelian if both $G_1$ and $G_2$ are abelian.

For notational simplicity, write

$(a,b)+(c,d)=(a+c,b+d)$

As long as it is understood that the additions in $G_1$ and $G_2$ may not be the same binary operations. (Sec. $3.4,\#27,$

Sec. $3.5,\#14,15,27,28,$

Sec. $3.6,\#12,$

Sec. $5.1,\#51$)

Sec. $3.4,\#27\ll$

Prove or disprove that each of the following groups with addition as defined in Exercises $52$ of section $3.1$ is cyclic.

a. ${\mathbb{Z}}_2\times {\mathbb{Z}}_3$

b. ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$

Sec. $3.5,\#14,15,27,28,$

Consider the additive group $ℝ$ of real numbers. Prove or disprove that each of the following mappings $\varnothing :ℝ\times ℝ\to ℝ\times ℝ$ is an automorphism. Equality and addition are defined on $ℝ\times ℝ$ in Exercise $52$ of section $3.1$.

a. $\varnothing (x,y)=(y,x)$

b. $\varnothing (x,y)=(-x,-y)$

Consider the additive group $ℝ$ of real numbers. Prove or disprove that each of the following mappings $\varnothing :ℝ\times ℝ\to ℝ$ is an isomorphism.

a. $\varnothing (x,y)=x$

b. $\varnothing (x,y)=x+y$

Consider the additive groups ${\mathbb{Z}}_2$, ${\mathbb{Z}}_3$, and ${\mathbb{Z}}_6$. Prove that ${\mathbb{Z}}_6$ is isomorphic to ${\mathbb{Z}}_2\times {\mathbb{Z}}_3$.

Let $G_1$, $G_2$, $H_1$, and $H_2$ be groups with respect to addition. If $G_1$ is isomorphic to $H_1$ and $G_2$ is isomorphic to $H_2$, prove that $G_1\times G_2$ is isomorphic to $H_1\times H_2$.

Sec. $3.6,\#12\ll$

Consider the additive group $ℝ$ of real numbers. Let $\varphi$ be a mapping from $ℝ\times ℝ$ to $ℝ$, where equality and addition are defined in Exercise 52 of Section 3.1. Prove or disprove that each of the following mappings $\varphi$ is a homomorphism. If $\varphi$ is a homomorphism, find ker $\varphi$, and decide whether $\varphi$ is an epimorphism or a monomorphism.

a. $\varphi (x,y)=x-y$

b. $\varphi (x,y)=2x$

Sec. $5.1,\#51\ll$

Let $R$ and $S$ be arbitrary rings. In the Cartesian product $R\times S$ of $R$ and $S$, define

$(r,s)=(r^{\prime },s^{\prime })$ if and only if $r=r^{\prime }$ and $s=s^{\prime }$

$(r_1,s_1)+(r_2,s_2)=(r_1+r_2,s_1+s_2)$,

$(r_1,s_1)\cdot (r_2,s_2)=(r_1{r}_2,s_1{s}_2)$.

a. Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of $R$ and $S$ and is denoted by $R\oplus S$.

b. Prove that $R\oplus S$ is commutative if both $R$ and $S$ are commutative.

c. Prove that $R\oplus S$ has a unity element if both $R$ and $S$ have unity elements.

d. Give an example of rings $R$ and $S$ such that $R\oplus S$ does not have a unity element.