RINGS WITH A FINITE SET OF NONNILPOTENTS

Let R be a ring and let N denote the set of nilpotent elements of R. Let n be a nonnegative integer. The ring R is called a θn-ring if the number of elements in R which are not in N is at most n. The following theorem is proved: If R is a θn-ring, then R is nil or R is finite. Conversely, if R is a nil ring or a finite ring, then R is a θn-ring for some n. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.

Let R be a ring and let N denote the set of nilpotent elements of R. Let n be a nonnegative integer.The ring R is called a 0 -ring if the number of elements n in R which are not in N is at most n; that is, IR/NI < n.One question which suggests itself is the following: what are necessary and sufficient conditions for a given ring R to be a 0 -ring?The answer is given in the following.n 2. MAIN RESULTS.THEOREM I.If R is a -ring, then R is nil or R is finite.Conversely, if n R is a nil ring or a finite ring, then R is a -ring for some integer n. n In the proof of Theorem I, we use the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case.Finally, we consider the general case.
In preparation for the proof of Theorem I, we first prove the following lemmas.
LEMMA I. Any subring and any homomorphic image of a 8 -ring is a -ring.n n This follows at once from the definition of a 0 -ring.n LEMMA 2. Let RI,R2,...,Rn+ 1 be rings where each R i has an identity.Then the direct sum R 1 R 2 Rn+1 is not a -ring.n PROOF.Let u I (I,0,0,...,0), u 2 (0,I,0,...,0), Un+I (0,0, 0,I), where each u.. is an element of the direct sum R 1 -R 2 + Rn+1.Then u i 'N for each i 1,2,...,n + I, where N denotes the set of nilpotent elements of R. Hence Rn+1 is not a 8-ring.n LEMMA 3. If D is a division ring which is also a 8 -ring, then D is finite.
n This follows at once from the difinition of a 8-ring.n LEMMA 4. If R is a primitive ring which is also a 8 -ring, then R is finite.n In fact, R is isomorphic to a complete matrix ring D over a finite division ring D. m PROOF.Suppose R D for any division ring D and any positive integer m. m (We shall show that this leads to a contradiction.)Then, by Jacobson's Density Theorem [I; p. 33], for each positive integer q, there exists a subrlng R 0 of R which maps homomorphlcally onto D for some division ring D. In particular, there q exists a subring R I of R which maps homomorphically onto Dn+I.Hence, by Lemma I, Dn+l is a 0 -ring.
(2.1) n Now, define AI,A2,...,An+ I to be the following (n + I) (n + I) matrices in Dn+l: A. has zero entries in all rows except row i, but A. has entries of I throughout row i (i 1,2,...,n + I).
It is readily verified that each A. is idempotent and thus each A i is not nilpotent.l Hence, IDn+I\NI > n, where N denotes the set of nilpotent elements of Dn+I.There- fore, Dn+1 is not a 8n-ring, which contradicts (2.1).This contradiction shows that R D for some division ring D and some positive integer m.Now, since the division m ring D is a subring of the 8n-ring Dm, D is finite, by Lemmas 1 and 3. Thus, R is finite, and the proof is complete.
LEMMA 5.If R is a semisimple ring which is also a 8 -ring, then R is finite.n PROOF.By contradiction.Thus, suppose that R is a semisimple ring which is also a 8 -ring, and suppose R is not finite.Since R is semisimple, there exists n ideal I (eel) of R such that [I; p. 14] e I--(0)', each R/Ie is primitive (2.2) By Lemma I, R/I is a 8 -ring.Hence, by (2.2) and Lemma 4, Hence, using Lemma I, E" R/I is a 8 -ring.This, however, contradicts Lemma 2 n i=l s 1 (see (2.3)).This contradiction shows that R is finite, and the lemma is proved.
We are now in a position to prove our main result, stated at the beginning.
PROOF of Theorem I. Let R be a 8n-ring, and let J denote the Jacobson radical of R. We claim that J is nil.We prove this by contradiction.Thus, suppose J is not nil.Then, for some a, a e J, a N, where N denotes the set of nilpotent elements of R. Let 2 3 n+l S {a,a ,a ,...,a }, (aJ, aN) Since a N, a i N for each i 1,2 ,n + I, and hence S = R\N.Therefore, ISI < IR\NI < n (2.6) i aj since R is a 8 -ring.In view of (2.5) and (2.6), we see that a for some n k distinct positive integers i, j, and hence some power of a is idempotent, say a k k Thus, a is an idempotent element of J, and hence, as is well known, a 0.
Therefore, a e N, a contradiction.This contradiction proves that J is nil.Now, if J R, then R is nil.So suppose J # R. Then R/J is a semisimple ring with more than one element.Moreover, by Lemma i, R/J is a 8 -ring, and hence n by Lemma 5, R/J is finite.We have thus shown that R/J is a finite semisimple ring with more than one element, and hence, as is well known, R/J has an idenitity.Let u be the identity element of R/J, and let N + u {n + u neN}; N is the set of nilpotents of R. (2.7) We claim that For, if a e (N + u) n N, then a N and a n + u for some n N. Hence a n + u an invertible element of R/J.Thus, a is an invertible element of R/J, which contradicts the hypothesis that a e N (and hence a is nilpotent).This contradiction proves (2.8).In view of (2.8), we have N + u c_q R\N, and hence IN + u < IR\NI <_ n (2.9) since R is a 8 n-ring.But INI IN + ul and hence by (2.9) N is finite.The net result, then, is that both N and R\N are finite, and hence R is finite.We have thus shown that, in any case, R is nil (if J R) or R is finite (if J # R).
The converse part of Theorem 1 is trivial.

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a Dm R/I is finite, for any e (2.3), we see that R is finite, a contradiction.This contradiction k