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 Commutativity and structure of rings with commuting nilpotents

1983 ◽  
Vol 6 (1) ◽  
pp. 119-124
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Commutative Rings ◽  
Structure Theory ◽  
Nilpotent Elements

LetRbe a ring and letNdenote the set of nilpotent elements ofR. LetZdenote the center ofR. Suppose that (i)Nis commutative, (ii) for everyxinRthere existsx′ϵ<x>such thatx−x2x′ϵN, where<x>denotes the subring generated byx, (iii) for everyx,yinR, there exists an integern=n(x,y)≥1such that both(xy)n−(yx)nand(xy)n+1−(yx)n+1belong toZ. ThenRis commutative and, in fact,Ris isomorphic to a subdirect sum of nil commutative rings and local commutative rings. It is further shown that both conditions in hypothesis (iii) are essential. The proof uses the structure theory of rings along with some earlier results of the authors.

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.

Small potential counterexamples to the pure semisimplicity conjecture

2018 ◽  
Vol 17 (10) ◽  
pp. 1850183
Author(s):  
José L. García
Keyword(s):  
Division Ring ◽  
Continued Fraction ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Representation Type ◽  
Ring Extension ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Triangular Matrix Ring ◽  
Semisimple Ring

The pure semisimplicity conjecture or pssc states that every left pure semisimple ring has finite representation type. Let [Formula: see text] be division rings, and assume we identify conditions on a [Formula: see text]-[Formula: see text]-bimodule [Formula: see text] which are sufficient to make the triangular matrix ring [Formula: see text] into a left pure semisimple ring which is not of finite representation type. It is then said that those conditions yield a potential counterexample to the pssc. Simson [17–20] gave several such conditions in terms of the sequence of the left dimensions of the left dual bimodules of [Formula: see text]. In this paper, conditions with the same purpose are given in terms of the continued fraction attached to [Formula: see text], and also through arithmetical properties of a division ring extension [Formula: see text].

scholarly journals On transposes of nilpotent matrices over arbitrary rings

1977 ◽  
Vol 23 (3) ◽  
pp. 366-370 ◽  
Author(s):  
Thomas P. Kezlan
Keyword(s):  
Division Ring ◽  
Polynomial Identity ◽  
Jacobson Radical ◽  
Commutator Ideal ◽  
Nilpotent Elements ◽  
Nilpotent Matrices

AbstractIt is shown that if every nilpotent 2 × 2 matrix over a ring has nilpotent transpose, then the commutator ideal must be contained in the Jacobson radical, thus generalizing a result of R. S. Gupta, who considered the division ring case. Moreover, if the nilpotent elements form an ideal or if the ring satisfies a polynomial identity, then the above property of the transpose implies that in fact the commutator ideal must be nil.

DEFINABLE SUBCATEGORIES OVER PURE SEMISIMPLE RINGS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250099 ◽  
Author(s):  
NGUYEN VIET DUNG ◽  
JOSÉ LUIS GARCÍA
Keyword(s):  
Module Category ◽  
Indecomposable Modules ◽  
Functor Category ◽  
The Family ◽  
Finite Set ◽  
Semisimple Ring

Let R be a right pure semisimple ring, and [Formula: see text] be a family of sources of left almost split morphisms in mod-R. We study the definable subcategory [Formula: see text] in Mod-R determined by the family [Formula: see text], and show that [Formula: see text] has several nice properties similar to those of the category Mod-R. For example, its functor category [Formula: see text] is a module category, and preinjective objects of [Formula: see text] are sources of left almost split morphisms in [Formula: see text] and in mod-R. As an application, it is shown that if R is a right pure semisimple ring with no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in mod-R (in particular, if R is right pure semisimple hereditary), then any definable subcategory of Mod-R determined by a finite set of indecomposable right R-modules contains only finitely many non-isomorphic indecomposable modules.

The inclusion-exclusion principle for finitely many isolated sets

1986 ◽  
Vol 51 (2) ◽  
pp. 435-447
Author(s):  
J. C. E. Dekker
Keyword(s):  
Nonnegative Integer ◽  
Natural Extension ◽  
Exclusion Principle ◽  
Finite Set ◽  
Equivalence Type ◽  
Image Position

AbstractA nonnegative integer is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of α, ⊂ for inclusion and ⊂+ for proper inclusion. Let α, β1,…, βk be subsets of some set υ. Then α′ stands for υ−α and β1 … βk for β1 ∩ … ∩ βk. For subsets α1, …, αr of υ we write:Note that α0 = υ, hence s0 = N(υ). If the set υ is finite, the classical inclusion-exclusion principle (abbreviated IEP) statesIn this paper we generalize (a) and(b) to the case where α1, …, αr are subsets of some countable but isolated set υ. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalizations of (a) and (b) are proved in §3. Since they involve recursive distinctness, this notion is discussed in §2. In §4 we consider a natural extension of “the sum of the elements of a finite set σ” to the case where σ is countable. §5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection Λ of all isols which permit us to further generalize IEP by substituting μ(α) for Req α.

scholarly journals A generalization of a theorem of Wedderburn

1973 ◽  
Vol 8 (2) ◽  
pp. 181-185 ◽  
Author(s):  
Steve Ligh
Keyword(s):  
Division Ring ◽  
Commutative Rings ◽  
Left Identity ◽  
Nilpotent Elements

Outcalt and Yaqub have extended the Wedderburn Theorem which states that a finite division ring is a field to the case where R is a ring with identity in which every element is either nilpotent or a unit. In this paper we generalize their result to the case where R has a left identity and the set of nilpotent elements is an ideal. We also construct a class of non-commutative rings showing that our generalization of Outcalt and Yaqub's result is real.

scholarly journals Isomorphism between Endomorphism Rings of Modules over A Semisimple Ring

2020 ◽  
Vol 26 (2) ◽  
pp. 170-174
Author(s):  
Hery Susanto ◽  
Santi Irawati ◽  
Indriati Nurul Hidayah ◽  
Irawati
Keyword(s):  
Division Ring ◽  
Artinian Ring ◽  
Endomorphism Rings ◽  
Simple Ring ◽  
Semisimple Ring

Our question is what ring R which all modules over R are determined, up to isomorphism, by their endomorphism rings? Examples of this ring are division ring and simple Artinian ring. Any semi simple ring does not satisfy this property. We construct a semi simple ring R but R is not a simple Artinian ring which all modules over R are determined, up to isomorphism, by their endomorphism rings.

NILPOTENT ELEMENTS AND NIL-ARMENDARIZ PROPERTY OF MONOID RINGS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250080 ◽  
Author(s):  
M. HABIBI ◽  
A. MOUSSAVI
Keyword(s):  
Infinite Order ◽  
Monoid Ring ◽  
Nilpotent Elements ◽  
Strongly Connected ◽  
Armendariz Rings ◽  
Nil Ring

Antoine [Nilpotent elements and Armendariz rings, J. Algebra 319(8) (2008) 3128–3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. For a monoid M, we introduce nil-Armendariz rings relative to M, which is a generalization of nil-Armendariz rings and we investigate their properties. This condition is strongly connected to the question of whether or not a monoid ring R[M] over a nil ring R is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc.7 (1956) 35–48]. This is true for any 2-primal ring R and u.p.-monoid M. If the set of nilpotent elements of a ring R forms an ideal, then R is nil-Armendariz relative to any u.p.-monoid M. Also, for any monoid M with an element of infinite order, M-Armendariz rings are nil M-Armendariz. When R is a 2-primal ring, then R[x] and R[x, x-1] are nil-Armendariz relative to any u.p.-monoid M, and we have nil (R[M]) = nil (R)[M].

scholarly journals JUST NON COMMUTATIVE VARIETIES OF OPERATOR ALGEBRAS AND RINGS WITH SOME CONDITIONS ON NILPOTENT ELEMENTS

1996 ◽  
Vol 27 (1) ◽  
pp. 59-65
Author(s):  
YURI N. MAL'CEV
Keyword(s):  
Operator Algebras ◽  
Finite Ring ◽  
Nilpotent Elements ◽  
Engel Identity

In §1 it is given a classification of Just noncommutative varieties of associative over algebras over commutative Jacobson ring with unity. In [1], [4] are given different proofs of the commutativity of a finite ring with central nilpotent elements. In §2 we give generalizations of these results for infinite rings and for the case of Engel identity.

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