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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 ideal; that is, $rL \subseteq L$ for all $r \in R$. Now let $a \in L$ be nonzero. Since $R$ is a division ring, there exists $b \in R$ such that $ba = 1 \in L$. Then for all $r \in R$, $r \cdot 1 \in L$, and in fact $L = R$. Thus the only left ideals of $R$ are $0$ and $R$.

($\Leftarrow$) Suppose the only left ideals of $R$ are $0$ and $R$, and let $a \in R$ be nonzero. Now the left ideal $Ra$ generated by $a$ is nonzero since it contains $a$, so that $Ra = R$. So there exists an element $b \in R$ such that $ba = 1$. Similarly, there exists $c \in R$ such that $cb = 1$. Now $$c = c(ba)=(cb)a=a,$$hence $ab = 1$. So $a$ is $a$ unit in $R$, and thus $R$ is a division ring.


Linearity

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