let be a non-commutative ring, and define to be the set of zero-divisors of the ring . Suppose that , for any . Prove that is an ideal of .
Unfortunately, hatsoff's argument only works in the commutative case. In a noncommutative ring, in general.
However, it is true that if then . If you multiply on the left by or on the right by , you get 0. Provided that both and are nonzero, that says that and hence . Therefore (and that result trivially also holds if or is 0).
A similar argument, with and interchanged, shows that . Together with the previous result, that gives . But then , so .
That shows that D(R) is closed under addition. For the other property of an ideal, if and then clearly . But I cannot see any reason why (in a nocommutative ring) there should exist a nonzero element c such that . All I can conclude is that in the ring R, the set of left zero-divisors is a right ideal, and the set of right zero-divisors is a left ideal.
thanks very much , you have already proved it ,as you have showed if and then which means that and this shows that is a right ideal . we also have which shows that is a left ideal and therefore is an ideal. (note that here is the set of left and right zero-divisors together and not just the two-sided ones.)