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.

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

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.

scholarly journals On distributively generated near-rings

1969 ◽  
Vol 16 (3) ◽  
pp. 239-243 ◽  
Author(s):  
Steve Lich
Keyword(s):  
Division Ring ◽  
Nonzero Element ◽  
Ring Theory ◽  
Left Identity

The following theorems in ring theory are well-known:1. Let R be a ring. If e is a unique left identity, then e is also a right identity.2. If R is a ring with more than one element such that aR = R for every nonzero element a ε R, then R is a division ring.3. A ring R with identity e ≠ 0 is a division ring if and only if it has no proper right ideals.

FINITARY AUTOMORPHISM GROUPS OVER NON-COMMUTATIVE RINGS

2001 ◽  
Vol 64 (3) ◽  
pp. 611-623 ◽  
Author(s):  
B. A. F. WEHRFRITZ
Keyword(s):  
Division Ring ◽  
Automorphism Groups ◽  
Commutative Rings ◽  
Linear Groups ◽  
Locally Finite ◽  
Arbitrary Ring ◽  
Finitary Automorphism

The notion of a group of finitary automorphisms of an arbitrary module over an arbitrary ring is introduced, and it is shown how properties of such groups can be derived from the case where the ring is a division ring (that is, from the properties of finitary skew linear groups). The results are particularly strong if either the group is locally finite or the module is Noetherian.

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.

A Decomposition of Rings Generated by Faithful Cyclic Modules

1989 ◽  
Vol 32 (3) ◽  
pp. 333-339 ◽  
Author(s):  
Gary F. Birkenmeier
Keyword(s):  
Decomposition Theorem ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Nilpotent Elements ◽  
Strongly Regular ◽  
Regular Rings

AbstractA ring R is said to be generated by faithful right cyclics (right finitely pseudo-Frobenius), denoted by GFC (FPF), if every faithful cyclic (finitely generated) right R-module generates the category of right R-modules. The class of right GFC rings includes right FPF rings, commutative rings (thus every ring has a GFC subring - its center), strongly regular rings, and continuous regular rings of bounded index. Our main results are: (1) a decomposition of a semi-prime quasi-Baer right GFC ring (e.g., a semiprime right FPF ring) is achieved by considering the set of nilpotent elements and the centrality of idempotnents; (2) a generalization of S. Page's decomposition theorem for a right FPF ring.

scholarly journals Nilpotents and units in skew polynomial rings over commutative rings

1979 ◽  
Vol 28 (4) ◽  
pp. 423-426 ◽  
Author(s):  
M. Rimmer ◽  
K. R. Pearson
Keyword(s):  
Commutative Ring ◽  
Finite Order ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

AbstractLet R be a commutative ring with an automorphism ∞ of finite order n. An element f of the skew polynomial ring R[x, α] is nilpotent if and only if all coefficients of fn are nilpotent. (The case n = 1 is the well-known description of the nilpotent elements of the ordinary polynomial ring R[x].) A characterization of the units in R[x, α] is also given.

A Theorem on Rings

1953 ◽  
Vol 5 ◽  
pp. 238-241 ◽  
Author(s):  
I. N. Herstein
Keyword(s):  
Commutative Rings ◽  
Nilpotent Elements

In a recent paper, Kaplansky [2] proved the following theorem: Let R be a ring with centre Z, and such that xn(x) ∈ Z for every x∈ R. If R, in addition, is semi-simple then it is also commutativeThe existence of non-commutative rings in which every element is nilpotent rules out the possibility of extending this result to all rings. One might hope, however, that if R is such that xn(x) ∈ Z for all x ∈ R and the nilpotent elements of R are reasonably “well-behaved,” then Kaplansky's theorem should be true without the restriction of semi-simplicity.

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