Von Neumann regular ring
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A Certain Conditions on Some Rings Give P.P.Ring Open
Many new results were obtained in this paper about P.P. ring. Semi-primitive ring R with (dcc) on principal-ideal P is always P.P.ring. Also, St-G-P.P. ring R is given to answer new question; is B-rings are P.P. ring. Also regular and Von …
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GENERALIZING $\pi$-REGULAR RINGS Open
We introduce the class of weakly nil clean rings, as rings $R$ in which for\nevery $a \\in R$ there existan idempotent $e$ and a nilpotent $q$ such that $a-e-q\n\\in eRa$. Every weakly nil clean ring is exchange. Weakly nil clean rings con…
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A CLASSIFICATION OF RING ELEMENTS IN SKEW PBW EXTENSIONS OVER COMPATIBLE RINGS Open
For a skew PBW extension over a right duo compatible ring, we characterize several kinds of their elements such as units, idempotent, von Neumann regular, $\\pi$-regular and the clean elements. As a consequence of our treatment, we extend …
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Gaussian Trivial Ring Extensions and FQP-rings Open
Let A be a commutative ring and E a non-zero A-module. Necessary and\nsufficient conditions are given for the trivial ring extension R of A by E to\nbe either arithmetical or Gaussian. The possibility for R to be B{\\'e}zout is\nalso studi…
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Quasi regular modules and trivial extension Open
Recall that a ring $R\ $is said to be a quasi regular ring if its total quotient ring $q(R)\ $is \textit{von Neumann regular}. It is well known that a ring $R\ $is quasi regular if and only if it is a reduced ring satisfying the property: …
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On <i>S</i> -2-absorbing submodules and vn-regular modules Open
Let R be a commutative ring and M an R -module. In this article, we introduce the concept of S -2-absorbing submodule. Suppose that S ⊆ R is a multiplicatively closed subset of R. A submodule P of M with ( P : R M ) ∩ S = ∅ is said to be a…
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On Nonnil-S-Noetherian Rings Open
Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties o…
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On the Commutative Rings with At Most Two Proper Subrings Open
The commutative rings with exactly two proper (unital) subrings are characterized. An initial step involves the description of the commutative rings having only one proper subring.
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Lamplighter groups and von Neumann‘s continuous regular ring Open
Let be a discrete group. Following Linnell and Schick one can define a continuous ring associated with . They proved that if the Atiyah Conjecture holds for a torsion-free group , then is a skew field. Also, if has torsion and the Stro…
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Invo-regular unital rings Open
It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. M…
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On weakly clean and weakly exchange rings having the strong property Open
We define two classes of rings calling them weakly clean rings and weakly exchange rings both equipped with the strong property. Although the classes of weakly clean rings and weakly exchange rings are different, their two proper subclasse…
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Chromatic Number and some Properties of Pseudo-Von Neumann Regular graph of Cartesian Product of Rings Open
Let R be a commutative ring, the Pseudo – Von Neumann regular graph of the ring R is define as a graph whose vertex set consists of all elements of R and any two distinct vertices a and b are adjacent if and only if , this graph is denoted…
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Some Results on Skew Generalized Power Series Rings Open
Let $R$ be a ring, $(S,\\leq)$ a strictly ordered monoid and $\\omega \\colon S \\to \\operatorname{End}(R)$ a monoid homomorphism. The skew generalized power series ring $R[[S,\\omega]]$ is a common generalization of (skew) polynomial rin…
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MacWilliams’ extension theorem for infinite rings Open
Finite Frobenius rings have been characterized as precisely those finite\nrings satisfying the MacWilliams extension property, by work of Wood. In the\npresent note we offer a generalization of this remarkable result to the realm\nof Artin…
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Semicommutativity of rings by the way of idempotents Open
In this paper, we focus on the semicommutative property of rings via idempotent elements. In this direction, we introduce a class of rings, so-called right e-semicommutative rings. The notion of right e-semicommutative rings generalizes th…
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Mass Formula for Self-Orthogonal and Self-Dual Codes over Non-Unital Rings of Order Four Open
We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring I=a,b|2a=2b=0,a2=b,ab=0 and the noncommutative ring E=a,b|2a=2b=0,a2=a,b2=b,ab=a,ba=b. We use these structu…
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Some Extensions of Generalized Morphic Rings and EM-rings Open
Let R be a commutative ring with unity. The main objective of this article is to study the relationships between PP-rings, generalized morphic rings and EM-rings. Although PP-rings are included in the later rings, the converse is not in ge…
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Regular rings and their properties Open
In this paper, we present some properties of regular rings. The properties are developed from the properties of regular semigroups. The properties of regular rings here are especially related to the center of the rings, division rings, Boo…
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ZERO DIVISOR GRAPHS OF SKEW GENERALIZED POWER SERIES RINGS Open
Let R be a ring, (S,${\leq}$) a strictly ordered monoid and ${\omega}$ : S ${\rightarrow}$ End(R) a monoid homomorphism. The skew generalized power series ring R[[S,${\omega}$]] is a common generalization of (skew) polynomial rings, (skew)…
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λ-rings, Φ-λ-rings, and Φ-Δ-rings Open
Let R be a commutative ring with unity. The notion of ?-rings, ?-?-rings, and ?-?-rings is introduced which generalize the concept of ?-domains and ?-domains. A ring R is said to be a ?-ring if the set of all overrings of R is linearly ord…
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Group-regular rings Open
We propose different generalizations of unit-regularity of elements in general rings (non necessarily unital rings). We then study general rings for which all elements have these properties. We notably compare them with unit-regular ideals…
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Characterizing $S$-flat modules and $S$-von Neumann regular rings by uniformity Open
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $T$ is called $u$-$S$-torsion ($u$- always abbreviates uniformly) provided that $sT=0$ for some $s\in S$. The notion of $u$-$S$-exact sequences is also introduced from…
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On regular ideals in reduced rings Open
Let R be a commutative ring with identity and X be a Tychonoff space. An ideal I of R is Von Neumann regular (briefly, regular) if for every a ? I, there exists b ? R such that a = a2b. In the present paper, we obtain the general form of a…
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RINGS WITH MANY REGULAR ELEMENTS Open
In this paper we introduce rings that satisfy regular 1-stable range. These rings are left-right symmetric and are generalizations of unit 1-stable range. We investigate characterizations of these kind of rings and show that these rings ar…
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Noncommutative G-semihereditary rings Open
In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [For…
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The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings Open
Let M_n (R_1 [[S_1,≤_1,ω_1]]) and M_n (R_2 [[S_2,≤_2,ω_2]]) be a matrix rings over skew generalized power series rings, where R_1,R_2 are commutative rings with an identity element, (S_1,≤_1 ),(S_2,≤_2 ) are strictly ordered monoids, ω_1:S…
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On flat epimorphisms of rings and pointwise localizations Open
We prove some new results on flat epimorphisms of commutative rings and pointwise localizations. Especially among them, it is proved that a ring $R$ is an absolutely flat (von-Neumann regular) ring if and only if it is isomorphic to the po…
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GENERALIZATION OF VON-NEUMANN REGULAR RINGS TO VON-NEUMANN REGULAR MODULES Open
An element r in a commutative ring R is called regular if there exist s∈R such that rsr=r. Ring R is called vN (von-Neumann)-regular ring if every element is regular. Recall that for any ring R always can be considered as module over itsel…
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ON JACOBSON AND NIL RADICALS RELATED TO POLYNOMIAL RINGS Open
This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus th…
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Some commutative ring extensions defined by almost Bézout condition Open
In this paper, we study the almost Bézout property in different commutative ring extensions, namely, in bi-amalgamated algebras and pairs of rings. In Section 2, we deal with almost Bézout domains issued from bi-amalgamations. Our results …