Nonzero Nilpotent Ideals Research Articles

Nilpotent Ideals in Generalized Amenable Banach Algebras

In this article, some properties such as the approximation property of non-zero nilpotent ideals in generalized amenable Banach algebras are investigated. Also, we show that in pseudo-contractible Banach algebras, non-zero nilpotent ideals cannot have the approximation property.

ON LEAVITT PATH ALGEBRAS OVER COMMUTATIVE RINGS

In this article, basic ideals in a Leavitt path algebra over a com- mutative unital ring are studied. It is shown that for a nite acyclic graph E and a commutative unital ring R, the Leavitt path algebra LR(E) is a direct sum of minimal basic ideals and that for a commutative ring R and a graph E satisfying Condition (L), the Leavitt path algebra LR(E) has no non-zero nilpotent basic ideals. Uniqueness theorems for Leavitt path algebras over commutative unital rings are also discussed.

On the radical of a free Malcev algebra

We prove that the prime radical radM of the free Malcev algebra M of rank more than two over a field of characteristic = 2 coincides with the set of all universally Engelian elements of M. Moreover, let T (M) be the ideal of M consisting of all stable identities of the split simple 7-dimensional Malcev algebra M over F . It is proved that radM = J(M) ∩ T (M), where J(M) is the Jacobian ideal of M. Similar results were proved by I. Shestakov and E. Zelmanov for free alternative and free Jordan algebras. An algebra M is called a Malcev algebra if it satisfies the identities x = 0, J(x, xy, z) = J(x, y, z)x, where J(x, y, z) = (xy)z + (zx)y + (yz)x is the Jacobian of the elements x, y, z [7, 9, 5]. Since for a Lie algebra the Jacobian of any three elements vanishes, Lie algebras fall into the variety of Malcev algebras. Among the non–Lie Malcev algebras, the traceless elements of the octonion algebra with the product given by the commutator [x, y] = xy − yx is one of the most important examples [9, 5, 6]. In 1977 I. P. Shestakov [11] proved that the free Malcev algebra Mn on n ≥ 9 free generators is not semiprime; that is, Mn contains nonzero nilpotent ideals. In 1979, V.T. Filippov [3] extended this result to free Malcev algebras with more than four generators. Therefore, the prime radical radMn = 0 for n > 4, and a natural question on the description of this radical arises. For free alternative algebras, it was proved by Shestakov in [10] that the prime radical coincides with the set of nilpotent elements. A similar fact was established by E. Zel′manov [15] for free Jordan algebras. In anticommutative algebras, the role of nilpotent elements is played by engelian elements. An element a of an algebra M is called engelian if the operator of right multiplication Ra : x → xa is nilpotent. We will call an element a ∈ M universally engelian if, for every algebra M ′ ⊇ M , the element a is engelian in M ′. In other words, the image Ra of the element a in the (associative) universal multiplicative enveloping algebra R(M) of M is nilpotent. In the present paper, we prove that the Received by the editors February 23, 2011 and, in revised form, March 31, 2011. 2010 Mathematics Subject Classification. Primary 17D10, 17D05, 17A50, 17A65.

Rings Whose Nonzero Modules Have Maximal Submodules

All rings are assumed to be associative and (except for nil-rings and some stipulated cases) to have nonzero identity elements. Expressions such as a “Noetherian ring” mean that the corresponding right and left conditions hold. A module is said to be simple if it does not have nonzero proper submodules. A submodule N of a module M is called a maximal submodule (in M) if the factor module M/N is simple. A ring A is called a right max ring (or a right Bass ring) if every nonzero right A-module has a maximal submodule. A ring is said to be orthogonally finite if it does not contain an infinite set of nonzero orthogonal idempotents. Bass [11] proved that a ring with the minimum condition on principal left ideals is an orthogonally finite right max ring. Cozzens [25] and Koifman [61] constructed examples of orthogonally finite right max rings which do not satisfy the minimum condition on principal left ideals. This review contains some new and some old results related to max rings. We present the necessary notation and definitions. Let A be a ring. We denote the center of the ring A by C(A). A subset B of A is said to be central (in A) if B ⊆ C(A). If M and N are two modules, then Hom(M,N) denotes the Abelian group formed by all homomorphisms M → N . For a module M , we denote by End(M) the endomorphism ring Hom(M,M) of M . For a subset B of A, the right annihilator and the left annihilator of B in A are denoted by rA(B) and A(B), respectively. We can omit the subscripts if the situation is obvious. An element a of A is said to be right regular (resp. left regular) in A if r(a) = 0 (resp. (a) = 0). A ring A is said to be regular if for every element a of A, there exists an element b of A such that a = aba. A ring A is said to be right weakly regular if B2 = B for every right ideal B of A. A ring without nonzero nilpotent ideals is called a semiprime ring. A ring is called a prime ring if the product of any two of its nonzero ideals is not equal to zero. A ring A is said to be simple if each of its nonzero ideals coincides with A. A ring is called a domain if the product of any two of its nonzero elements is not equal to zero. For a ring A, an ideal P of A is called a prime (resp. semiprime and completely prime) ideal if the factor ring A/P is a prime ring (resp. a semiprime ring and a domain). A ring A is called a P.I.-ring (or a ring with polynomial identity) if A satisfies the polynomial identity f(x1, . . . , xn) = 0, where f(x1, . . . , xn) is a polynomial in noncommutative variables with coefficients in the ring of integers Z, and Z coincides with its ideal generated by the coefficients of f(x1, . . . , xn). A submodule N of a module M is said to be essential (in M) if N has nonzero intersection with any nonzero submodule of the module M . In this case, we say that M is an essential extension of the module N . A submodule N of a module M is called a superfluous submodule (in M) if N +M ′ = M for every proper submodule M ′ of M .

The approximation property and nilpotent ideals in amenable Banach algebras

It is well known that non-zero nilpotent ideals in amenable Banach algebras must be infinite-dimensional. We show that under certain additional hypotheses such ideals cannot even have the approximation property.

NILPOTENCY, SOLVABILITY AND RADICALS IN CATEGORIES

Abstract Nilpotent and solvable ideals are defined and investigated in categories. The relation between the prime radical and the sum of the solvable ideals (which is also a radical) is discussed in categories. For example: If an object satisfies the maximal condition for ideals, then the prime radical is equal to the sum of the solvable ideals. Certain generalizations of theorems in rings, groups, Lie algebras, etc. are also proven, for example: An ideal α: I → A is semiprime if and only if A/I contains no non-zero nilpotent ideals.

Semi-prime Bernstein algebras

In this paper we study the semiprime case. That is, we consider Bernstein algebras that do not have nonzero nilpotent ideals of index two. We prove that any such algebra is Jordan. Furthermore, under the condition that the algebra is finitely generated, we show that it must be a field. The proofs require characteristic different from two. Our work implies that nearly all (finitely generated) Bernstein algebras possess nonzero ideals which are nilpotent of index two. The only ones which do not are the fields.

A NIL RADICAL FOR Ω-GROUPS

Abstract In this paper we define a nil radical for fl-groups which has the property that an ideal P of an fl-group G is a semi-prime ideal if and only if G/P has no non-zero nilpotent ideals. This generalises the case for rings to Ω-groups.

On rings whose number of centralizers of ideals is finite

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I∼J if and only ifV R (I)=V R(J), whereV R() is the centralizer inR. LetI R=F/∼. Then we can see thatn(I R), the cardinality ofI R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st−ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I R)=m for any given natural numberm.

An upper radical property and an answer to a problem of L.C.A. van Leeuwen and G.A.P. Heyman

1. It is well known that the Baer's lower radical property B, coincides with the lower radical property determined by the class of all nilpotent rings, and also with the upper radical property determined by the class of all semiprime rings, that is, rings without nonzero nilpotent ideals (N. J. Dwp~srcY [5]). L. C. A. VAN LF.EUW~N and G.A.P. HEYMAN [10] in an almost recent paper generalizing the notion of nilpotent rings introduced the class of almost nilpotent rings, that is, of rings every proper homomorphic image of which, is nilpotent ring,* and proved analogously, that the lower radical property L~ determined by the class of all almost nilpotent rings coincides with the upper radical property determined by the class of all rings without nonzero almost nilpotent ideals and it holds B~L2~Bq, where B~0 is the Andrunakiev 6's antisimple radical property (V. A. ANDRUNAKIEVI~ [1], [2]). Evidently, for every almost nilpotent ring R it holds R~ where ~o is the first limit ordinal number, and R~ ('1 R". n(N Generalizing further the notion of nilpotent rings, the ring R is said to be trausfinitely nilpotent if there exists an ordinal number r such that Rr=0. We recall that the power R' is defined as follows: If r is a nonlimit ordinal number then Rr=R "-I.R while if r is a limit ordinal number then Rr= ORbb<r The transfinitely nilpotent property was first discussed by R. BAER [3] and thirty years later F. A. Sz2{sz [15] asked whether the lower radical property determined by the class of all transfinitely nilpotent rings is special, and attributed an upper bound to this class. B. J. GARDNER [6] proved that this lower radical property is trivial because it coincides with the class of all rings, rejecting so the existence of an upper bound and restricting its interest. It is now natural to ask: Which is the upper rddical property ~ determined by the class M of all rings without nonzero transfinitely nilpotent ideals? In the present paper this radical property is studied. It is proved that the class M is weakly special and that B~ T<=B~,. Further, T~J and J~ T, where J is the Jacobson's radical class. Finally, this paper answers a problem raised by L.C.A. van Leeuwen and G.A.P. Heyman, proving B<L~ ~:<T<--B~, by an example which is the Jacobson radical of a Noetherian principal left ideal domain

Semiprime semigroup rings and a problem of J. Weissglass

If R is a ring and S is a semigroup, the corresponding semigroup ring is denoted by R[S]. A ring is semiprime if it has no nonzero nilpotent ideals. A semigroup S is a semilattice P of semigroups Sα if there exists a homomorphism φ of S onto the semilattice P such that Sα = αφ−1 for each α ∈ P.

Semiprime semigroup rings and a problem of J. Weissglass

If R is a ring and S is a semigroup, the corresponding semigroup ring is denoted by R[S]. A ring is semiprime if it has no nonzero nilpotent ideals. A semigroup S is a semilattice P of semigroups Sα if there exists a homomorphism φ of S onto the semilattice P such that Sα = αφ−1 for each α ∈ P.

Remarks on the commutativity of rings

Introduction. A celebrated theorem of N. Jacobson [7] asserts that if (1) Xn(z) =x for every x in a ring R, where n(x) is an integer greater than one, then R is commutative. In a recent paper [2], I. N. Herstein has shown that it is enough to require that (1) holds for those x in R which are commutators: x= [y, z] =yz-zy of two elements of R. The purpose of this note is to show that if R has no nonzero nilpotent ideals, we may restrict x in (1) to iterated commutators of any fixed degree. We also obtain a weaker result for arbitrary rings. An important tool in the proof of our results is a lemma which generalizes a result of Kaplansky [4] to the effect that the only elements of a primitive ring which commute with all commutators are in the center. This tool is also useful in extending and complementing some results of Divinsky [1] on commuting isomorphisms of simple rings. These extensions include some recent results of Posner [9] on derivations in prime rings. As a final remark, we indicate an exceedingly brief proof of Herstein's result [3 ] on Jordan derivations in prime rings. Although the proof of our first result follows that of Herstein very closely, we present a self-contained account for the convenience of the reader.

An embedding of 𝑃𝐼-rings

minimal identity of a simple algebra. We prove that for a PI-ring which has no nonzero nilpotent ideals, the standard identity Sd(X) =0, where d is an even integer, is the unique (up to a numerical factor) minimal identity which is linear in each of its indeterminates. The term standard identity was ascribed in [2] to the polynomial identity: