TWO-SIDED ESSENTIAL NILPOTENCE

An ideal I of a ring A is essentially nilpotent if I contains a nilpotent ideal N of A such that J 91N # 0 whenever J is a nonzero ideal of A contained in I. We show that each ring A has a unique largest essentially nilpotent ideal EN(A). We study the properties of EN(A) and, in particular, we investigate how this ideal behaves with respect to related rings.

Let A be a ring and suppose I A. If K A and K _ I then K is A-essential in I if 0 # B , A and B C: I imply that B f3 K # 0. The ideal I is an essentially nilpotent ideal of A if there is a nilpotent ideal N of A such that N C_ I and N is A-essential in I.We shall denote the prime radical of A by N(A).Recall that N(A) is the intersection of the prime ideals of A and that if I , A, then N(I) 1 91N(A).
Essential nilpotence was first studied by Fisher  [2].In this paper we show that each ring A contains a unique largest essentially nilpotent ideal which we denote by EN(A).We establish various results concerning this ideal and, in particular, we investigate how this ideal behaves with respect to related rings.For example, we show that EN(R[x])= EN(R)[x] and that if G is a finite group of automorphisms of R and R has no G I-torsion, then EN(R.G) EN(R).G.Proposition 1.Let I A. The following are equivalent.
1.I is an essentially nilpotent ideal of A; 2. A has an ideal Z such that Z 2 O, Z C I and Z is A-essential in I; 3. If 0 K A and K C I, then K contains a nonzero nilpotent ideal of A; and 4. N(I) is A-essential in I. PROOF. 1 implies 2. This follows as in [2, Lemma 2.1], but we repeat the argument for the convenience of the reader.Let {Z" , E A} be the collection of all ideals J of A such that j2= 0 and J _C 1.Let O= {F C_ A" E{Z," E F} is direct}.Using Zorn's lemma we may choose M maximal inO.Let Z=E{ZA'M}.ThenZC_IandZ2=0.LetB,A, BC_I.IfB0then B N K # 0 where K is a nilpotent ideal of A, K I. Thus B f3 K and hence B contains a nonzero ideal J of A such that j2 0. The maximality of M ensures that Z f3 J # 0 and so Z is A-essential in I.
3 implies 4. If J is a nilpotent ideal of A and J _C I, then J C_ N(I) so this implication is also clear. 4implies 1.Since every nonzero ideal of A contained in N(I) contains an ideal J of A with j2= 0, the argument in the proof that implies 2 shows that there is an ideal Z of A, Z2= 0, Z C_ N(I) and Z is A-essential in N(I).Since m(I) A and m(I)is A-essential in I it follows that Z is A-essential in I.
Let I and J be essentially nilpotent ideals of the ring A. If 0 # K A and K _C I + J, then either 0 # KI C_ I or 0 # KJ C_ J or K 2 0. In any case, K contains a nonzero nilpotent ideal of A and so I + J is essentially nilpotent by 3 of the above Proposition.A similar argument shows that the sum of all the essentially nilpotent ideals of A is essentially nilpotent.This unique largest essentially nilpotent ideal of A will be denoted by EN(A).
Proposition 2.1./f 0 is an automorphism o.f A, then O(EN(A))= EN(A). 2. For any ring A,A/EN(A) is semiprime.In particular, if A B, then EN(A),: B. 3. If I A, EN(I) I EN(A). 4. /f 0 e e 2 E A, then EY(eAe) C_ eEY(A)e. 5.If A has an identity, 0 e e 2 A and AeA A, then Eg(eAe) eEN(A)e.
PROOF. 1. is clear.For the proof of 2. suppose EN(A)C_ JnA and J2C EN(A).If 0 g ,: A,K C_ J then g 2 0 implies g C_ N(A) C_ BY(A) and g 2 # 0 implies K 2 K2gl EN(A) # O. Hence EN(A) is A-essential in J and so J is essentially nilpotent.Hence J EN(A) and the proof of 2. is complete.
For the proof of 3. we begin by showing that EN(A)f'l I is an essentially nilpotent ideal of I. Let 0 J I,J CC_ EN(A)f'l I.In view of 3 of Proposition it is enough to show that J contains a nonzero nilpotent ideal of I.If J is itself nilpotent this is certainly the case.If J is not nilpotent, j.3 # 0 where J* is the ideal of A which is generated by J. Since (j.)3 _C EN(A), (5*) 3 contains a nonzero nilpotent ideal of A and since (J*)3 C_ J by Andrunakievic's Lemma this completes the proof that EN(A) fl I CC_ EN(I).
From 2. we know that EN(I)A and it follows immediately from 4 in Proposition 1 that EN(I) is an essentially nilpotent ideal of A.
To prove 5 it suffices to show that eEN(A)e C_ EN(eAe), and to do this it is enough to show that eEY(A)e is an essentially nilpotent ideal of eAe.Now Y(eEY(A)e)= eN(EN(A))e eg(A)e and we will show that eg(A)e is eAe-essential in eEN(A)e.Let 0 # W eAe, W C eEN(A)e.Let W* denote the ideal of A which is generated by W. Since W'C_ EN(A), K W* tDN(A)# O. Also, eKe C W fl eN(A)e so the proof will be complete if we show that eKe # 0. If eKe 0, then AKA AeAKAeA C_ AeKeA O.But since A has an identity and K # O, AKA # O.
If R and S are rings with the same identity and R C_ S, then S is a free normalizing extension of R and S is a free right and left R-module with a basis X such that xR Rx for all x fi X.Note that in this case each x X determines an automorphism 0 x of R defined by xOx(r rx for all r E R. A free normalizing extension S of R satisfies the essential condition if whenever U C_ V are ideals of S with U S-essential in V and I , R such that IV O, then 1V fl U O. If S is a free centralizing eztension of R; that is, 0 x is the identity automorphism for all z E X, then certainly S satisfies the essential condition because in this case IV ,S.Also, if G is a finite group of automorphisms of R and R has no G ]-torsion, then the crossed product R,G satisfies the essential condition.This is because a minor modification of the proof of Lemma 1.2 (ii) in Passman  [3] shows that if U and V are ideals of R,G with U R,G-essential in V, then U is essential as an R-R*G subbimodule of V. THEOREM 3. If S is a free normalizing eztension of R which satisfies the essential condition and is such that N(S)= N(R)S, then EN(S)= EN(R)S.
PROOF.We first show that EN(R)SC_EN(S).Since EN(R) is invariant under automorphisms of R, EN(R)S is an ideal of S. We show that N(S) is S-essential in EN(R)S.Let 0 # T S,T C_ EN(R)S and denote the normalizing basis of S over R by X {z A" A fi A}.Choose 0 # v E{aAz A" A A} in T where a A EN(R) and so that v has a minimal number of coefficients not in N(nR ).Suppose ,5 A and a,5 .N (R).Since 0 # Ra,bR C_ EN(R) there are zj, yj R such that 0 # zja,byj N(R).Then j=l w Z xjv#,5(yj)A# =' akz$ + Z xja,bx,5#,5(yj)A='E,5 akzA + Z zja,byjx,5 j=l j=l j=l where the a are elements of R with the property that aN(R)if aeN(R).
Sic jj # 0, "----1 W J and since w has fewer coefficients not in N(R) than does v we have reached a contradiction.
It follows that vN(R)S=N(S) and hence N(S)is S-essential in EN(R)S.Hence EN(R)S C_ EN(S).
Suppose that 0 v EN(S), v .E N(R)S.Let v y {ax" A} and assume ,5 A is such that a _ EN(R).Then N(a)is not R-essential in (a)+ N(a)where (a) denotes the ideal of R which is generated by a. Hence there is an ideal I of R, 0 I C_ (a) + N(R) and I n N(R) 0. It follows that IEN(S)fl N(R)S 0 because if {r,x," , .A,r, _ R} is in IEN(S) then r, E I for all .Since I is not nilpotent and IN(R) C I flN(R)=O, Ia$ O. Hence Iv # 0 and so IEN(S) 7/:O.Since N(S) is S-essential in EN(S) and S satisfies the essential condition, IEN(S) Cl N(S) 7/: O.This contradicts our previous conclusion that IEN(S) Cl N(R)S 0 because N(S) N(R)S.Hence EN(S) C_ EN(R)S.
It is well-known that if S is a finite normalizing extension of R, then N(S)::) N(R) and so it follows from the proof of the theorem that if S is a finite free normalizing extension of R, then EN(S) D_ EN(R).
COROLLARY 4. If Mn(A) denotes the ring of n n matrices with entries from A, then EN(Mn(A)) Mn(EN(A)).
PROOF.First assume that A has an identity.Since Mn(A) is a free centralizing extension of If A does not have an identity, let A' be the usual (Dorroh) unital extension of A. Then from 3 of Proposition 2, EN(Mn(A)) Mn(A) fl EN(Mn(A')) Mn(A f3 Mn(EN(A')) Mn(A f3 EN(A')) =Mn(EN(A)).
COROLLARY 5.If G is a finite group of automorphisms of A and A has no G[-torsion, then EN(A,G)= EN(A).G where A.G is the crossed product.PROOF.As in Corollary 4 we may assume that A has an identity, and it follows from [3, Theorem 2.2] that N(A.G)= N(A).G so the theorem applies.PROOF.As above we may assume that A has an identity and [1, Lemma 2L] shows that N(A[x]) N(A)[x].So, since A[x] is a free centralizing extension of A, the theorem applies.COROLLARY 7. If R and S are rings with identity which are Morita equivalent, then R is essentially nilpotent if and only if S is essentially nilpotent.PROOF.This follows immediately from 5 of Proposition 2 and Corollary 4. COROLLARY 8. Let R be a ring with identity and let G be a finite group of automorphisms of R such that GI/s invertible in R. Then EN(RG) C_ EN(R).
PROOF.Let e GI-I' g.Then e is idempotent in the skew group ring R,G and e(R,G)e=RGe-RG.Since EN(RGe)=EN(RG)e it follows that EN(RG) c_ EN(R).
We note that EN(R) R does not in general imply that EN(RG) # O.For example, let where Q is the rational field.The cyclic group G {e, ct} of order 2 acts as automorphisms of R via and c 0 c 0 c (A)) Mn(N(A)) it follows from the theorem that EN(Mn(A)) EN(A)Mn(A)= Mn(EN(A)).