The Existence of a Simple Nil Ring (After Agata Smoktunowicz)

2003 ◽  
pp. 129-214
Author(s):  
I. V. L’vov
Keyword(s):  
Nil Ring

scholarly journals A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350121 ◽  
Author(s):  
AGATA SMOKTUNOWICZ
Keyword(s):  
Jacobson Radical ◽  
Analogous Result ◽  
Graded Rings ◽  
Radical Ring ◽  
Graded Ring ◽  
Nil Ring ◽  
Nil Radical

It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil radicals, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper, it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

2011 ◽  
Vol 21 (05) ◽  
pp. 745-762 ◽  
Author(s):  
TAI KEUN KWAK ◽  
YANG LEE
Keyword(s):  
Polynomial Ring ◽  
Regular Ring ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Nilpotent Elements ◽  
Von Neumann Regular ◽  
Armendariz Rings ◽  
Ring Extensions ◽  
Nil Ring

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

Some Simple Properties of Simple Nil Rings

1966 ◽  
Vol 9 (2) ◽  
pp. 197-200 ◽  
Author(s):  
W. A. McWorter
Keyword(s):  
Unsolved Problem ◽  
Finitely Generated ◽  
Nilpotent Ring ◽  
Locally Nilpotent ◽  
Nil Ring

An outstanding unsolved problem in the theory of rings is the existence or non-existence of a simple nil ring. Such a ring cannot be locally nilpotent as is well known [ 5 ]. Hence, if a simple nil ring were to exist, it would follow that there exists a finitely generated nil ring which is not nilpotent. This seemed an unlikely situation until the appearance of Golod's paper [ 1 ] where a finitely generated, non-nilpotent ring is constructed for any d ≥ 2 generators over any field.

scholarly journals Rings decomposed into direct sums ofJ-rings and nil rings

1985 ◽  
Vol 8 (1) ◽  
pp. 205-207 ◽  
Author(s):  
Hisao Tominaga
Keyword(s):  
Direct Consequence ◽  
Direct Sums ◽  
Direct Sum ◽  
Nil Ring

LetRbe a ring (not necessarily with identity) and letEdenote the set of idempotents ofR. We prove thatRis a direct sum of aJ-ring (every element is a power of itself) and a nil ring if and only ifRis stronglyπ-regular andEis contained in someJ-ideal ofR. As a direct consequence of this result, the main theorem of [1] follows.

scholarly journals On Primitive Ideals in Graded Rings

2008 ◽  
Vol 51 (3) ◽  
pp. 460-466 ◽  
Author(s):  
Agata Smoktunowicz
Keyword(s):  
Prime Ideal ◽  
Jacobson Radical ◽  
Graded Rings ◽  
Dimensional Algebra ◽  
Infinite Dimensional ◽  
Primitive Ideals ◽  
Nil Ring

AbstractLet R = be a graded nil ring. It is shown that primitive ideals in R are homogeneous. Let A = be a graded non-PI just-infinite dimensional algebra and let I be a prime ideal in A. It is shown that either I = ﹛0﹜ or I = A. Moreover, A is either primitive or Jacobson radical.

scholarly journals Generating Adjoint Groups

2019 ◽  
Vol 62 (3) ◽  
pp. 733-738 ◽  
Author(s):  
Be'eri Greenfeld
Keyword(s):  
Open Problem ◽  
Jacobson Radical ◽  
The Other ◽  
Adjoint Group ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nil Ring

AbstractWe prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.

scholarly journals Some conditions for finiteness of a ring

1988 ◽  
Vol 11 (2) ◽  
pp. 239-242 ◽  
Author(s):  
Howard E. Bell
Keyword(s):  
Chain Condition ◽  
Zero Divisors ◽  
Periodic Ring ◽  
Descending Chain Condition ◽  
Nil Ring ◽  
Ascending Chain Condition

Extending a result of Putcha and Yaqub, we prove that a non-nil ring must be finite if it has both ascending chain condition and descending chain condition on non-nil subrings. We also prove that a periodic ring with only finitely many non-central zero divisors must be either finite or commutative.

scholarly journals Rings with a finite set of nonnilpotents

1979 ◽  
Vol 2 (1) ◽  
pp. 121-126 ◽  
Author(s):  
Mohan S. Putcha ◽  
Adil Yaqub
Keyword(s):  
Division Ring ◽  
Nonnegative Integer ◽  
Finite Ring ◽  
Structure Theory ◽  
Nilpotent Elements ◽  
Finite Set ◽  
Semisimple Ring ◽  
Nil Ring ◽  
Primitive Ring

LetRbe a ring and letNdenote the set of nilpotent elements ofR. Letnbe a nonnegative integer. The ringRis called aθn-ring if the number of elements inRwhich are not inNis at mostn. The following theorem is proved: IfRis aθn-ring, thenRis nil orRis finite. Conversely, ifRis a nil ring or a finite ring, thenRis aθn-ring for somen. 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.

scholarly journals On periodic rings

2001 ◽  
Vol 25 (6) ◽  
pp. 417-420
Author(s):  
Xiankun Du ◽  
Qi Yi
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sum ◽  
Periodic Rings ◽  
Necessary And Sufficient ◽  
Nil Ring ◽  
Positive Integers

It is proved that a ring is periodic if and only if, for any elementsxandy, there exist positive integersk,l,m, andnwith eitherk≠morl≠n, depending onxandy, for whichxkyl=xmyn. Necessary and sufficient conditions are established for a ring to be a direct sum of a nil ring and aJ-ring.

RADICALS WITH THE α-AMITSUR PROPERTY

2008 ◽  
Vol 07 (03) ◽  
pp. 347-361 ◽  
Author(s):  
SODNOMKHORLOO TUMURBAT ◽  
ROBERT WISBAUER
Keyword(s):  
Jacobson Radical ◽  
Radical Class ◽  
Prime Rings ◽  
Nil Ring

A radical γ of rings is said to have the Amitsur property if for all rings A, γ(A[X]) = (γ(A[X]) ∩ A)[X]. Let Xα denote a set of indeterminates of cardinality α. We say that γ has the α-Amitsur property if for all rings A, γ(A[Xα]) = (γ(A[Xα]) ∩ A)[Xα]. We study properties of this type of radicals and show relationships with other known radicals for rings. A ring A is said to be an absolute γ-ring if A[x1,…, xn] ∈ γ, for any n ∈ ℕ. We show that A is an absolute 𝔾-ring for the Brown–McCoy radical 𝔾, if and only if A is in the radical class S determined by the unitary strongly prime rings. Moreover, A is an absolute nil ring if and only if A is an absolute J-ring, where J denotes the Jacobson radical.

Sign in / Sign up

Export Citation Format

Share Document