scholarly journals On S-2-absorbing submodules and vn-regular modules

2020 ◽  
Vol 28 (2) ◽  
pp. 239-257
Author(s):  
Gülşen Ulucak ◽  
Ünsal Tekir ◽  
Suat Koç
Keyword(s):  
Commutative Ring ◽  
Closed Subset ◽  
Von Neumann ◽  
Von Neumann Regular

AbstractLet 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 an S-2-absorbing submodule if there exists an element s ∈ S and whenever abm ∈ P for some a, b ∈ R and m ∈ M, then sab ∈ (P :R M) or sam ∈ P or sbm ∈ P. Many examples, characterizations and properties of S-2-absorbing submodules are given. Moreover, we use them to characterize von Neumann regular modules in the sense [9].

scholarly journals On zero-dimensionality and fragmented rings

1999 ◽  
Vol 60 (1) ◽  
pp. 137-151
Author(s):  
Jim Coykendall ◽  
David E. Dobbs ◽  
Bernadette Mullins
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Krull Dimension ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Finite Products ◽  
Regular Rings

A commutative ring R is said to be fragmented if each nonunit of R is divisible by all positive integral powers of some corresponding nonunit of R. It is shown that each fragmented ring which contains a nonunit non-zero-divisor has (Krull) dimension ∞. We consider the interplay between fragmented rings and both the atomic and the antimatter rings. After developing some results concerning idempotents and nilpotents in fragmented rings, along with some relevant examples, we use the “fragmented” and “locally fragmented” concepts to obtain new characterisations of zero-dimensional rings, von Neumann regular rings, finite products of fields, and fields.

When is every module with essential socle a direct sum of automorphism-invariant modules?

2019 ◽  
Vol 19 (10) ◽  
pp. 2050185
Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Saboura Dolati Pish Hesari ◽  
Mehdi Khoramdel
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Essential Extension ◽  
Principal Ideal Ring ◽  
Von Neumann ◽  
Simple Modules ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Ideal Ring ◽  
Von Neumann Regular

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

scholarly journals A note on regular modules

1974 ◽  
Vol 11 (3) ◽  
pp. 359-364 ◽  
Author(s):  
V.S. Ramamurthi
Keyword(s):  
Commutative Ring ◽  
Projective Modules ◽  
Von Neumann ◽  
Von Neumann Regular

Kaplansky's observation, namely, a commutative ring R is (von Neumann) regular if and only if each simple R-module is injective, is generalized to projective modules over a commutative ring.

Some algebraic and homological properties of a family of quotients of the Rees algebra

2019 ◽  
Vol 18 (12) ◽  
pp. 1950232
Author(s):  
Mahnaz Salek ◽  
Elham Tavasoli ◽  
Abolfazl Tehranian ◽  
Maryam Salimi
Keyword(s):  
Commutative Ring ◽  
Regular Ring ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Ideal Formula ◽  
Von Neumann Regular ◽  
Semisimple Ring ◽  
The Ideal

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a proper ideal of [Formula: see text]. In this paper, we study some algebraic and homological properties of a family of rings [Formula: see text], with [Formula: see text], that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. Specially, we study when [Formula: see text] is a von Neumann regular ring, a semisimple ring and a Gaussian ring. Also, we study the classical global and weak global dimensions of [Formula: see text]. Finally, we investigate some homological properties of [Formula: see text]-modules and we show that [Formula: see text] and [Formula: see text] are Gorenstein projective [Formula: see text]-modules, provided some special conditions.

scholarly journals Note on integral closures of semigroup rings

2000 ◽  
Vol 31 (2) ◽  
pp. 137-144
Author(s):  
Ryuki Matsuda
Keyword(s):  
Commutative Ring ◽  
Quotient Group ◽  
Integral Domain ◽  
Regular Ring ◽  
Quotient Ring ◽  
Closed Domain ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Integrally Closed

Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

scholarly journals On regular ideals in reduced rings

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3715-3726 ◽  
Author(s):  
A.R. Aliabady ◽  
R. Mohamadian ◽  
S. Nazari
Keyword(s):  
Closed Subset ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Closed Set ◽  
Von Neumann ◽  
Arbitrary Ring ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Isolated Points ◽  
Maximal Regular Ideal

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 regular ideal in C(X) which is OA, for some closed subset A of ?X, for which Ac?X ? (P(X))?, where P(X) is the set of all P-points of X. We show that the ideals and subrings such as CK(X), C?(X), C?(X), SocmC(X) and M?X\X are regular if and only if they are equal to the socle of C(X). We carry further the study of the maximal regular ideal, for instance, it is shown that for a vast class of topological spaces (we call them OPD-spaces) the maximal regular ideal is OX\I(X), where I(X) is the set of isolated points of X. Also, for this class, the socle of C(X) is the maximal regular ideal if and only if I(X) contains no infinite closed set. We also show that C(X) contains an ideal which is both essential and regular if and only if (P(X))? is dense in X. Finally it is shown that, for semiprimitive rings pure ideals are of the form OA which A is a closed subset of Max(R), also a P-point of X = Max(R) is introduced and it is shown that the maximal regular ideal of an arbitrary ring R is OX\P(X), which P(X) is the set of P-points of X = Max(R).

Radical factorization for trivial extensions and amalgamated duplication rings

2020 ◽  
pp. 2150025
Author(s):  
Tiberiu Dumitrescu ◽  
Najib Mahdou ◽  
Youssef Zahir
Keyword(s):  
Commutative Ring ◽  
Regular Ring ◽  
Trivial Extension ◽  
Ring Extension ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Amalgamated Duplication

Let [Formula: see text] be a commutative ring extension such that [Formula: see text] is a trivial extension of [Formula: see text] (denoted by [Formula: see text]) or an amalgamated duplication of [Formula: see text] along some ideal of [Formula: see text] (denoted by [Formula: see text]. This paper examines the transfer of AM-ring, N-ring, SSP-ring and SP-ring between [Formula: see text] and [Formula: see text]. We study the transfer of those properties to trivial ring extension. Call a special SSP-ring an SSP-ring of the following type: it is the trivial extension of [Formula: see text] by a C-module [Formula: see text], where [Formula: see text] is an SSP-ring, [Formula: see text] a von Neumann regular ring and [Formula: see text] a multiplication C-module. We show that every SSP-ring with finitely many minimal primes which is a trivial extension is in fact special. Furthermore, we study the transfer of the above properties to amalgamated duplication along an ideal with some extra hypothesis. Our results allows us to construct nontrivial and original examples of rings satisfying the above properties.

GENERALIZATIONS OF COMPLEMENTED RINGS WITH APPLICATIONS TO RINGS OF FUNCTIONS

2009 ◽  
Vol 08 (01) ◽  
pp. 17-40 ◽  
Author(s):  
M. L. KNOX ◽  
R. LEVY ◽  
W. WM. MCGOVERN ◽  
J. SHAPIRO
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Regular Element ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Ring Of Quotients

It is well known that a commutative ring R is complemented (that is, given a ∈ R there exists b ∈ R such that ab = 0 and a + b is a regular element) if and only if the total ring of quotients of R is von Neumann regular. We consider generalizations of the notion of a complemented ring and their implications for the total ring of quotients. We then look at the specific case when the ring is a ring of continuous real-valued functions on a topological space.

Von Neumann Regular and Related Elements in Commutative Rings

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1017-1040 ◽  
Author(s):  
David F. Anderson ◽  
Ayman Badawi
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Von Neumann ◽  
Regular Elements ◽  
Zero Divisor Graph ◽  
Von Neumann Regular ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, π-regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph Γ(R) of R induced by the above elements.

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