# Answer to Question #24887 in Abstract Algebra for john.george.milnor

Question #24887
For any ring R and any ordinal &alpha;, define N&alpha;(R) as follows. For &alpha; = 1, N1(R) is a nil subideal of Nil*R which contains all nilpotent one-sided ideals of R. If &alpha; is the successor of an ordinal &beta;, define N&alpha;(R) = {r &isin; R : r + N&beta;(R) &isin; N1 (R/N&beta;(R))}. If &alpha; is a limit ordinal, define N&alpha;(R) = (Union over &beta;&lt;&alpha;) N&beta;(R). Show that Nil*R = N&alpha;(R) for any ordinal &alpha; with Card &alpha; &gt; Card R.
1
2013-02-27T05:48:39-0500
The N&alpha;(R)&rsquo;s form an ascending chainof ideals in Nil*R. Write P(R) = N&alpha;(R) where &alpha; is an ordinal with Card &alpha; &gt; Card R. Then, for any ordinal &alpha;&#039; with Card &alpha;&#039; &gt; Card R, we have P(R) = N&alpha;&#039; (R). Since P(R) &sube; Nil*R, it suffices to show that Nil*R &sube; P(R). Now R/P(R) has nononzero nilpotent ideals, so it is a semiprime ring. This means that P(R)is a semiprime ideal. Hence Nil*R &sube; P(R) since Nil*R is the smallest semiprime ideal of R.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!