Nilpotent local automorphisms of an independence algebra

1994 ◽  
Vol 124 (3) ◽  
pp. 423-436 ◽  
Author(s):  
Lucinda M. Lima
Keyword(s):  
Positive Integer ◽  
Inverse Monoid ◽  
Nilpotent Elements ◽  
Infinite Rank ◽  
Index 2

Let A be a strong independence algebra of infinite rank m. Let ℒ(A) be the inverse monoid of all local automorphisms of A. The Baer–Levi semigroup B over the algebra A is defined to be the subsemigroup of ℒ(A) consisting of all the elements α with dom α = A and corank im α = m. Let Km be the subsemigroup of ℒ(A) generated by B−1B. Then Km is inverse and it is generated (as a semigroup) by N2, the subset of ℒ(A) consisting of all nilpotent elements of index 2. The 2-nilpotent depth, Δ2(Km) of Km is defined to be the smallest positive integer t such that Km = N2∪…∪(N2)t. In fact, Δ2(Km) is either 2 or 3 and a criterion is found which distinguishes between the two cases.If N denotes the set of all nilpotents in ℒ(A), then the subsemigroup generated by N is also Km. In fact, Km is proved to be exactly N2.

scholarly journals Periodic rings with commuting nilpotents

1984 ◽  
Vol 7 (2) ◽  
pp. 403-406
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Fields ◽  
Boolean Ring ◽  
Fixed Integer ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Periodic Rings

LetRbe a ring (not necessarily with identity) and letNdenote the set of nilpotent elements ofR. Suppose that (i)Nis commutative, (ii) for everyxinR, there exists a positive integerk=k(x)and a polynomialf(λ)=fx(λ)with integer coefficients such thatxk=xk+1f(x), (iii) the setIn={x|xn=x}wherenis a fixed integer,n>1, is an ideal inR. ThenRis a subdirect sum of finite fields of at mostnelements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (takingn=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume thatInis a subring ofR.

scholarly journals A nilpotent-generated semigroup associated with a semigroup of full transformations

1988 ◽  
Vol 108 (1-2) ◽  
pp. 181-187 ◽  
Author(s):  
John M. Howie ◽  
M. Paula O. Marques-Smith
Keyword(s):  
Regular Cardinal ◽  
Principal Factor ◽  
Nilpotent Elements ◽  
Index 2 ◽  
Stable Elements ◽  
Regular Cardinality

SynopsisLet X be a set with infinite regular cardinality m and let ℱ(X) be the semigroup of all self-maps of X. The semigroup Qm of ‘balanced’ elements of ℱ(X) plays an important role in the study by Howie [3,5,6] of idempotent-generated subsemigroups of ℱ(X), as does the subset Sm of ‘stable’ elements, which is a subsemigroup of Qm if and only if m is a regular cardinal. The principal factor Pm of Qm, corresponding to the maximum ℱ-class Jm, contains Sm and has been shown in [7] to have a number of interesting properties.Let N2 be the set of all nilpotent elements of index 2 in Pm. Then the subsemigroup (N2) of Pm generated by N2 consists exactly of the elements in Pm/Sm. Moreover Pm/Sm has 2-nilpotent-depth 3, in the sense that

Ad-nilpotent Elements of Semiprime Rings with Involution

2018 ◽  
Vol 61 (2) ◽  
pp. 318-327
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Lie Algebra ◽  
Semiprime Ring ◽  
Extended Centroid ◽  
Nilpotent Elements ◽  
Semiprime Rings ◽  
Rings With Involution ◽  
Torsion Free

AbstractLet R be an n!-torsion free semiprime ring with involution * and with extended centroid C, where n > 1 is a positive integer. We characterize a ∊ K, the Lie algebra of skew elements in R, satisfying (ada)n = 0 on K. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if a, b ∊ R satisfy (ada)n = adb on R, where either n is even or b = 0, then (a − λ)[(n+1)/2] = 0 for some λ ∊ C.

scholarly journals Inverse semigroups generated by nilpotent transformations

1984 ◽  
Vol 99 (1-2) ◽  
pp. 153-162 ◽  
Author(s):  
John M. Howie ◽  
M. Paula O. Marques-Smith
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Nilpotent Elements ◽  
Inverse Subsemigroup ◽  
Symmetric Inverse Semigroup ◽  
Infinite Cardinality ◽  
Index 2 ◽  
Rees Quotient ◽  
The Ideal

SynopsisLet X be a set with infinite cardinality m and let B be the Baer-Levi semigroup, consisting of all one-one mappings a:X→X for which ∣X/Xα∣ = m. Let Km=<B 1B>, the inverse subsemigroup of the symmetric inverse semigroup ℐ(X) generated by all products β−γ, with β,γ∈B. Then Km = <N2>, where N2 is the subset of ℐ(X) consisting of all nilpotent elements of index 2. Moreover, Km has 2-nilpotent-depth 3, in the sense that Let Pm be the ideal {α∈Km: ∣dom α∣<m} in Km and let Lm be the Rees quotient Km/Pm. Then Lm is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image of Lm also has these properties and is congruence-free.

ON NIL SKEW ARMENDARIZ RINGS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250017 ◽  
Author(s):  
M. Habibi ◽  
A. Moussavi
Keyword(s):  
Positive Integer ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Rich Source ◽  
Ore Extension ◽  
Nilpotent Elements ◽  
Triangular Matrix Ring ◽  
Armendariz Rings ◽  
Type Property

Antoine [Nilpotent elements and Armendariz rings, J. Algebra319 (2008) 3128–3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. When the set of nilpotent elements of a ring R with an α-condition, namely α-compatibility, forms an ideal, we observe that R satisfies a nil Armendariz-type property, in the context of Ore extension R[x;α, δ]. For a 2-primal ring R with a derivation δ, R[x] is nil [Formula: see text]-skew Armendariz, and for a 2-primal ring R, R is nil α-skew Armendariz if and only if R[x] is nil [Formula: see text]-skew Armendariz, where α is an endomorphism of R with αk = id R, for some positive integer k. Moreover, we prove that a ring R is nil (α, δ)-skew Armendariz if and only if the n-by-n skew triangular matrix ring Tn(R, σ) is nil [Formula: see text]-skew Armendariz, for each endomorphism σ, with σ(1) = 1. A rich source of rings R, for which R[x] is nil [Formula: see text]-skew Armendariz, is provided.

scholarly journals On commutativity in certain rings

1970 ◽  
Vol 2 (1) ◽  
pp. 107-115 ◽  
Author(s):  
H.G. Moore
Keyword(s):  
Positive Integer ◽  
Polynomial Identity ◽  
Associative Ring ◽  
Arbitrary Element ◽  
Alternative Ring ◽  
Commutator Ideal ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Fixed Positive Integer ◽  
Associative Rings

I.N. Herstein has shown that an associative ring in which the nilpotent elements are “well-behaved”, and such that every element satisfies a certain polynomial identity, is commutative. This result is generalized here. Specifically, it is shown that an alternative ring R which satisfies the following three properties is commutative:(i) for x ∈ R, there exists an integer n(x) and a polynomial px (t) with integer coefficients such that xn+1p(x) = xn;(ii) for a fixed positive integer m, a a nilpotent and b an arbitrary element of R, a - am commutes with b - bm;(iii) for the same m, a and b, (ab+b)m = (ba+b)m and (ab)m = ambm.Examples are given to show that all three properties are essential, and it is shown that for associative rings certain modified versions of these properties are individually enough to assure that the commutator ideal of the ring is nil.

On quasi-radical of near-ring of polynomials

2019 ◽  
Vol 56 (2) ◽  
pp. 252-259
Author(s):  
Ebrahim Hashemi ◽  
Fatemeh Shokuhifar ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Nilpotent Elements ◽  
Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

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