A ring has only the trivial one-sided ideals precisely when it is a division ring

Let R be a ring with 1 \neq 0. Prove that R is a division ring if and only if the only left ideals of R are 0 and R. (The statement remains true when “left” is replaced by “right”.)


(\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 bab = b, so that cbab = cb, and hence ab = 1. So a is a unit in R, and thus R is a division ring.

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Comments

  • Abhishek Khetan  On October 17, 2011 at 11:26 pm

    have you assumed that R is commutative?
    I want to know how can we say that there exist b,c in R such that ba=ac=1 given that (a)=R.

    • nbloomf  On October 18, 2011 at 8:28 am

      Good question. I’ll have to think about this some more.

      • nbloomf  On October 18, 2011 at 10:59 am

        I think it’s fixed now. Thanks!

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