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.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

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.

A note on r-precious ring

2020 ◽  
pp. 2150231
Author(s):  
Nitin Bisht
Keyword(s):  
Regular Element ◽  
Nilpotent Element ◽  
Idempotent Element ◽  
Von Neumann ◽  
Regular Elements ◽  
Basic Properties ◽  
Euclidean Domain ◽  
Von Neumann Regular

An element of a ring [Formula: see text] is said to be [Formula: see text]-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring [Formula: see text] are [Formula: see text]-precious, then [Formula: see text] is called an [Formula: see text]-precious ring. We study some basic properties of [Formula: see text]-precious rings. We also characterize von Neumann regular elements in [Formula: see text] when [Formula: see text] is a Euclidean domain and by this argument, we produce elements that are [Formula: see text]-precious but either not [Formula: see text]-clean or not precious.

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.

A CLOSURE OPERATION IN RINGS

2001 ◽  
Vol 12 (07) ◽  
pp. 791-812
Author(s):  
PERE ARA ◽  
GERT K. PEDERSEN ◽  
FRANCESC PERERA
Keyword(s):  
Regular Element ◽  
Closure Operation ◽  
Von Neumann ◽  
Regular Elements ◽  
C Algebras ◽  
Partial Inverse ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Invertible Elements ◽  
Maximal Regular Ideal

We study the operation E → cl (E) defined on subsets E of a unital ring R, where x ∈ cl (E) if (x + Rb) ∩ E ≠ ∅ for each b in R such that Rx + Rb = R. This operation, which strongly resembles a closure, originates in algebraic K-theory. For any left ideal L we show that cl (L) equals the intersection of the maximal left ideals of R containing L. Moreover, cl (Re) = Re + rad (R) if e is an idempotent in R, and cl (I) = I for a two-sided ideal I precisely when I is semi-primitive in R (i.e. rad (R/I) = 0). We then explore a special class of von Neumann regular elements in R, called persistently regular and characterized by forming an "open" subset Rpr in R, i.e. cl (R\Rpr) = R\Rpr. In fact, R\Rpr = cl (R\Rr), so that Rpr is the "algebraic interior" of the set Rr of regular elements. We show that a regular element x with partial inverse y is persistently regular, if and only if the skew corner (1 - xy)R(1 - yx) is contained in Rr. If I reg (R) denotes the maximal regular ideal in R and [Formula: see text] the set of quasi-invertible elements, defined and studied in [4], we prove that [Formula: see text]. Specializing to C*-algebras we prove that cl (E) coincides with the norm closure of E, when E is one of the five interesting sets R-1, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], and that Rpr coincides with the topological interior of Rr. We also show that the operation cl respects boundedness, self-adjointness and positivity.

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.

Von Neumann regular graphs associated with rings

2018 ◽  
Vol 10 (03) ◽  
pp. 1850029
Author(s):  
Ali Jafari Taloukolaei ◽  
Shervin Sahebi
Keyword(s):  
Regular Graph ◽  
Regular Element ◽  
Regular Graphs ◽  
Von Neumann ◽  
Basic Properties ◽  
Von Neumann Regular ◽  
Nonzero Identity

Let [Formula: see text] be a ring with nonzero identity. By the Von Neumann regular graph of [Formula: see text], we mean the graph that its vertices are all elements of [Formula: see text] such that there is an edge between vertices [Formula: see text] if and only if [Formula: see text] is a Von Neumann regular element of [Formula: see text], denoted by [Formula: see text]. In this paper, the basic properties of [Formula: see text] are investigated and some characterization results regarding connectedness, diameter, girth and planarity of [Formula: see text] are given.

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).

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.

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