scholarly journals A Note on a Self Injective Ring

1965 ◽  
Vol 8 (1) ◽  
pp. 29-32 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Minimum Condition ◽  
Maximum Condition ◽  
Simple Ring ◽  
The Right ◽  
Primitive Ring

A ring R with unity is called right (left) self injective if the right (left) R-module R is injective [7]. The purpose of this note is to prove the following: Let R be a prime ring with a maximal annihilator right (left) ideal. If R is right (left) self injective then R is a primitive ring with a minimal one-sided ideal. If R satisfies the maximum condition on annihilator right (left) ideals and R is right (left) self injective then R is a simple ring with the minimum condition on one-sided ideals.

On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals

1971 ◽  
Vol 14 (3) ◽  
pp. 443-444 ◽  
Author(s):  
Kwangil Koh ◽  
A. C. Mewborn
Keyword(s):  
Prime Ring ◽  
Chain Condition ◽  
Prime Rings ◽  
Simple Ring ◽  
Descending Chain Condition ◽  
Image Position ◽  
The Right ◽  
Ascending Chain Condition

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such thatI is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

Rings with Enough Invertible Ideals

1983 ◽  
Vol 35 (1) ◽  
pp. 131-144 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Prime Ring ◽  
Regular Element ◽  
Elementary Proof ◽  
Identity Element ◽  
Quotient Ring ◽  
Minimum Condition ◽  
Prime Rings ◽  
Hereditary Noetherian Prime Ring ◽  
The Right

All rings are associative with identity element 1 and all modules are unital. A ring has enough invertible ideals if every ideal containing a regular element contains an invertible ideal. Lenagan [8, Theorem 3.3] has shown that right bounded hereditary Noetherian prime rings have enough invertible ideals. The proof is quite ingenious and involves the theory of cycles developed by Eisenbud and Robson in [5] and a theorem which shows that any ring S such that R ⊆ S ⊆ Q satisfies the right restricted minimum condition, where Q is the classical quotient ring of R. In Section 1 we give an elementary proof of Lenagan's theorem based on another result of Eisenbud and Robson, namely every ideal of a hereditary Noetherian prime ring can be expressed as the product of an invertible ideal and an eventually idempotent ideal (see [5, Theorem 4.2]). We also take the opportunity to weaken the conditions on the ring R.

On Compact Prime Rings and their Rings of Quotients

1968 ◽  
Vol 11 (4) ◽  
pp. 563-568 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Minimum Condition ◽  
Topological Ring ◽  
Prime Rings ◽  
Locally Compact ◽  
Simple Ring ◽  
Left Order ◽  
Rings Of Quotients ◽  
Right Order

In [10], it is defined that a right (or left) ideal I of a ring R is very large if the cardinality of R/I is finite. It is also proven in [10, Theorem 3.4] that if R is a prime ring with 1 such that its characteristic is zero, then R is a right order in a simple ring with the minimum condition on one sided ideals if every large right ideal of R is very large. In the present note, we shall prove that if R is a prime ring with 1 such that its characteristic is zero and R is also a compact topological ring, then R is a right and left order in a simple ring with the minimum condition on one sided ideals, which is also a non-discrete locally compact topological ring if and only if every large right ideal of R is open. In particular, if R is an integral domain with 1 (not necessarily commutative) such that its characteristic is zero, then R is openly embeddable [13, p. 58] in a locally compact (topological) division ring if and only if every large right ideal of R is open. Following S. Warner [13], we shall say R is openly embeddable in a quotient ring Q(R) if there is a topology on Q(R) which is compatible with its structure, which induces on R its given topology and for which R is an open subset.

m-potent commutators involving skew derivations and multilinear polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Ashraf ◽  
Sajad Ahmad Pary ◽  
Mohd Arif Raza
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Prime Rings ◽  
Skew Derivation ◽  
Martindale Quotient Ring ◽  
Multilinear Polynomials ◽  
The Right ◽  
The Relationship

AbstractLet {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

On Very Large One Sided Ideals of a Ring

1966 ◽  
Vol 9 (2) ◽  
pp. 191-196 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Cardinal Number ◽  
Singular Element

If R is a ring, a right (left) ideal of R is said to be large if it has non-zero intersection with each non-zero right (left) ideal of R [8]. If S is a set, let |S| be the cardinal number of S. We say a right (left) ideal I of a ring R is very large if |R/I| < < No. If a is an element of a ring R such that (a)r = {r ∊ R|ar = 0} is very large then we say a is very singular. The set of all very singular elements of a ring R is a two sided ideal of R. If R is a prime ring, then 0 is the only very singular element of R and a very large right (left) ideal of R is indeed large provided that R is not finite.

scholarly journals Unique factorisation rings

1992 ◽  
Vol 35 (2) ◽  
pp. 255-269 ◽  
Author(s):  
A. W. Chatters ◽  
M. P. Gilchrist ◽  
D. Wilson
Keyword(s):  
Prime Ideal ◽  
Prime Ring ◽  
Polynomial Identity ◽  
Noetherian Ring ◽  
Basic Theory ◽  
Maximal Order ◽  
Prime Element ◽  
Polynomial Extension ◽  
Simple Ring ◽  
Left And Right

Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

AN ENGEL CONDITION WITH AUTOMORPHISMS FOR LEFT IDEALS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350092 ◽  
Author(s):  
CHENG-KAI LIU
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Identity Automorphism ◽  
Semiprime Rings ◽  
Engel Condition ◽  
Positive Integers

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(xn0), xn1], xn2], …], xnk] = 0 for all x ∈ L, where n0, n1, n2, …, nk are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

scholarly journals On certain subrings of prime rings with derivations

1993 ◽  
Vol 54 (1) ◽  
pp. 133-141 ◽  
Author(s):  
M. Brešar ◽  
J. Vukman
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
General Result ◽  
Additional Assumption ◽  
Classical Theorem ◽  
Prime Rings

AbstractLet D be a nonzero derivation of a noncommutative prime ring R, and let U be the subring of R generated by all [D(x), x], x ∞ R. A classical theorem of Posner asserts that U is not contained in the center of R. Under the additional assumption that the characteristic of R is not 2, we prove a more general result stating that U contains a nonzero left ideal of R as well as a nonzero right ideal of R.

Rings with (x, y, z) – (z, y, x) in the right nucleus

2015 ◽  
Vol 08 (02) ◽  
pp. 1550023
Author(s):  
K. Jayalakshmi ◽  
G. Nageswari
Keyword(s):  
Prime Ring ◽  
Nonassociative Ring ◽  
The Right

Let R be a semiprime nonassociative ring satisfying (x, y, z)–(z, y, x) ∈ Nr then Nl = Nr where Nl and Nr are Lie ideals of R, the set {x ∈ Nr : (R, R, R)x = 0} = {x ∈ Nl : x(R, R, R) = 0} is an ideal of R, and it is contained in the nucleus. Further if [R, R]Nr ⊂ Nr and R is a prime ring with Nr ≠ 0 then R is either associative or commutative.

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