A GENERALIZATION OF PRÜFER'S ASCENT RESULT TO NORMAL PAIRS OF COMPLEMENTED RINGS

2011 ◽  
Vol 10 (06) ◽  
pp. 1351-1362 ◽  
Author(s):  
DAVID E. DOBBS ◽  
JAY SHAPIRO
Keyword(s):  
Regular Ring ◽  
Quotient Ring ◽  
Ring Extension ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Normal Pair ◽  
Reduced Ring ◽  
Von Neumann Regular ◽  
Integrally Closed ◽  
Torsion Free

Let R ⊆ T be a (unital) extension of (commutative) rings, such that the total quotient ring of R is a von Neumann regular ring and T is torsion-free as an R-module. Let T ⊆ B be a ring extension such that B is a reduced ring that is torsion-free as a T-module. Let R* (respectively, A) be the integral closure of R in T (respectively, in B). Then (R*, T) is a normal pair (i.e. S is integrally closed in T for each ring S such that R* ⊆ S ⊆ T) if and only if (A, AT) is a normal pair. This generalizes results of Prüfer and Heinzer on Prüfer domains to normal pairs of complemented rings.

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).

A note on modules over regular rings

1971 ◽  
Vol 4 (1) ◽  
pp. 57-62 ◽  
Author(s):  
K. M. Rangaswamy ◽  
N. Vanaja
Keyword(s):  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Free Left ◽  
Von Neumann Regular ◽  
Strongly Regular ◽  
Torsion Free ◽  
Regular Rings

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

Maximal Quotient Rings and S-Rings

1972 ◽  
Vol 24 (5) ◽  
pp. 835-850 ◽  
Author(s):  
E. P. Armendariz ◽  
Gary R. McDonald
Keyword(s):  
Left Ideal ◽  
Regular Ring ◽  
Quotient Ring ◽  
Injective Envelope ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Left Quotient ◽  
Von Neumann Regular ◽  
Singular Ideal ◽  
Quotient Rings

Throughout, we assume all rings are associative with identity and all modules are unitary. See [7] for undefined terms and [3] for all homological concepts.Let R be a ring, E(R) the injective envelope of RR, and H =HomR(E(R),E(R)). Then we obtain a bimodule RE(R)H. Let Q = HomH(E(R), E(R)). Q is called the maximal left quotient ring of R. Q has the property that if p, q ∈ Q, p ≠ 0, then there exists r ∈ R such that rp ≠ 0, rq ∈ R, i.e., Q is a ring of left quotients of R.A left ideal I of R is dense if for every x,y ∈ R,x ≠ 0, there exists r ∈ R such that rx ≠ 0, ry ∈ I. An alternate description of Q is Q = {x ∈ E(RR) : (R : x) is a dense left ideal of R{, where (R : x) = {r ∈ R : rx ∈ R}.The left singular ideal of R is Zl(R) = {r ∈ R : lR(r) is an essential left ideal of R}, where lR(r) = {x ∈ R : xr = 0}. If Zl(R) = (0), then Q is a left self-injective von Neumann regular ring [7, § 4.5]. Most of the previous work on maximal left quotient rings has been done in this case.

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.

Rings whose nilpotent elements form a Lie ideal

2014 ◽  
Vol 51 (2) ◽  
pp. 271-284 ◽  
Author(s):  
Yinchun Qu ◽  
Junchao Wei
Keyword(s):  
Regular Ring ◽  
Lie Ideal ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Nilpotent Elements ◽  
Reduced Ring ◽  
Von Neumann Regular ◽  
Strongly Regular

A ring R is called NLI (rings whose nilpotent elements form a Lie ideal) if for each a ∈ N(R) and b ∈ R, ab − ba ∈ N(R). Clearly, NI rings are NLI. In this note, many properties of NLI rings are studied. The main results we obtain are the following: (1) NLI rings are directly finite and left min-abel; (2) If R is a NLI ring, then (a) R is a strongly regular ring if and only if R is a Von Neumann regular ring; (b) R is (weakly) exchange if and only if R is (weakly) clean; (c) R is a reduced ring if and only if R is a n-regular ring; (3) If R is a NLI left MC2 ring whose singular simple left modules are Wnil-injective, then R is reduced.

scholarly journals RINGS OF DEFINABLE SCALARS OF VERMA MODULES

2007 ◽  
Vol 06 (05) ◽  
pp. 779-787 ◽  
Author(s):  
SONIA L'INNOCENTE ◽  
MIKE PREST
Keyword(s):  
Lie Algebra ◽  
Model Theory ◽  
Regular Ring ◽  
Verma Module ◽  
Closed Field ◽  
Verma Modules ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Canonical Map ◽  
Von Neumann Regular

Let M be a Verma module over the Lie algebra, sl 2(k), of trace zero 2 × 2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U( sl 2(k)) to RM is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules.

scholarly journals Commutative von Neumann regular rings are 1-Gröbner

2021 ◽  
pp. 30-30
Author(s):  
Zoran Petrovic ◽  
Maja Roslavcev
Keyword(s):  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Polynomials ◽  
Bner Basis ◽  
Regular Rings

Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

2011 ◽  
Vol 21 (05) ◽  
pp. 745-762 ◽  
Author(s):  
TAI KEUN KWAK ◽  
YANG LEE
Keyword(s):  
Polynomial Ring ◽  
Regular Ring ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Nilpotent Elements ◽  
Von Neumann Regular ◽  
Armendariz Rings ◽  
Ring Extensions ◽  
Nil Ring

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

Centres of Rank-Metric Completions

1985 ◽  
Vol 37 (6) ◽  
pp. 1134-1148
Author(s):  
David Handelman
Keyword(s):  
Regular Ring ◽  
Rank Function ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Rank Metric ◽  
Von Neumann Regular Rings ◽  
Completion Process ◽  
Von Neumann Regular ◽  
Metric Topology ◽  
Regular Rings

In this paper, we are primarily concerned with the behaviour of the centre with respect to the completion process for von Neumann regular rings at the pseudo-metric topology induced by a pseudo-rank function.Let R be a (von Neumann) regular ring, and N a pseudo-rank function (all terms left undefined here may be found in [6]). Then N induces a pseudo-metric topology on R, and the completion of R at this pseudo-metric, , is a right and left self-injective regular ring. Let Z( ) denote the centre of whatever ring is in the brackets. We are interested in the map .If R is simple, Z(R) is a field, so is discrete in the topology; yet Goodearl has constructed an example with Z(R) = R and Z(R) = C [5, 2.10]. There is thus no hope of a general density result.

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