Maximal Quotient Ring Research Articles

The Regular Ring and the Maximal Ring of Quotients of a Finite Baer *-Ring

The Regular Ring and the Maximal Ring of Quotients of a Finite Baer *-Ring

The regular ring and the maximal ring of quotients of a finite Baer *-ring

Necessary and sufficient conditions are obtained for extending the involution of a Baer ∗ \ast -ring to its maximal ring of quotients. Berberian’s construction of the regular ring of a Baer ∗ \ast -ring is generalized and this ring is identified (under suitable hypotheses) with the maximal ring of quotients.

When is the maximal ring of quotients projective?

Let R R be an associative ring with 1 1 , and Q Q its maximal ring of right quotients. If r r belongs to R R , a right insulator for r r in R R is a finite subset of R , { r i } i = 1 m R,\{ {r_i}\} _{i = 1}^m , such that the right annihilator of { r r i ; i = 1 , … , m } \{ r{r_i};i = 1, \ldots ,m\} is zero. Then we have: If Q Q is a projective right R R -module, Q Q is finitely generated; if R R is nonsingular, then Q Q is projective as a right R R -module if and only if there exists e = e 2 e = {e^2} in R R such that e R eR is injective and e e has a right insulator in R R ; under these circumstances, R = Q R = Q if and only if e e has a left insulator in R R . We prove some related results for torsionless Q Q , and give an example of a prime ring R R such that Q Q is a cyclic projective right R R -module, but R ≠ Q R \ne Q .

The regular ring and the maximal ring of quotients of a finite Baer $^{\ast} $-ring.

The regular ring and the maximal ring of quotients of a finite Baer $^{\ast} $-ring.

A Subdirect Decomposition of Semiprime Rings and its Application to Maximal Quotient Rings

Levy [2] has examined semiprime rings which are irredundant subdirect products of prime rings. In this note we look at the role of inessential prime ideals and see how every semiprime ring is a subdirect product of (i) a semiprime ring which is an irredundant subdirect product of prime rings, and (ii) a semiprime (nonprime) ring, all of whose prime ideals are essential. This leads to a direct sum decomposition of maximal left quotient rings of semiprime rings with left singular ideal zero.

Maximal Quotients of Semiprime PI-Algebras

J. Fisher [3] initiated the study of maximal quotient rings of semiprime PI-rings by noting that the singular ideal of any semiprime Pi-ring R is 0; hence there is a von Neumann regular maximal quotient ring $Q(R)$ of R. In this paper we characterize $Q(R)$ in terms of essential ideals of C = cent R. This permits immediate reduction of many facets of $Q(R)$ to the commutative case, yielding some new results and some rapid proofs of known results. Direct product decompositions of $Q(R)$ are given, and $Q(R)$ turns out to have an involution when R has an involution.

Maximal quotient rings of group rings

The two which have received the greatest attention are theclassical (Ore) quotient ring and the maximal (Utumi) quotient ring.The classical quotient ring has a relatively straightforward description,but it is only defined for rings which satisfy the so-called Ore con-dition. In contrast the maximal quotient ring is less easy to describebut is defined for all rings. In both cases there are distinct notionsof left and right quotient rings and we will always consider leftquotient rings.For group rings the classical quotient ring has been studied byHerstein and Small [2], Passman [5, 6], M. Smith [7], and P. F. Smith[81, and the maximal quotient ring has been studied by Burgess [1].This paper investigates the relationship of the maximal quotientrings of group rings, subgroup rings, and the centers of group rings.The object is to obtain for the maximal quotient ring analogues oftheorems of Passman and M. Smith on the classical quotient ring.Their techniques are used for the group ring arguments while thequotient ring arguments reflect the formalism of the maximal quotientring.

The Maximal Ring of Quotients of a Triangular Matrix Ring.

The Maximal Ring of Quotients of a Triangular Matrix Ring.

Regular Self-Injective Rings With a Polynomial Identity

This paper studies maximal quotient rings of semiprime P. I.-rings; such rings are regular, self-injective and satisfy a polynomial identity. We show that the center of a regular self-injective ring is regular self-injective; this enables us to establish that the center of the maximal quotient ring of a semiprime P. I.-ring R is the maximal quotient ring of the center of R, as well as some other relationships. We give two decompositions of a regular self-injective ring with a polynomial identity which enable us to show that such rings are biregular and are finitely generated projective modules over their center.

Maximal quotients of semiprime PI-algebras

J. Fisher [3] initiated the study of maximal quotient rings of semiprime PI-rings by noting that the singular ideal of any semiprime Pi-ring R is 0; hence there is a von Neumann regular maximal quotient ring Q ( R ) Q(R) of R. In this paper we characterize Q ( R ) Q(R) in terms of essential ideals of C = cent R. This permits immediate reduction of many facets of Q ( R ) Q(R) to the commutative case, yielding some new results and some rapid proofs of known results. Direct product decompositions of Q ( R ) Q(R) are given, and Q ( R ) Q(R) turns out to have an involution when R has an involution.

Prime regular rings of quotients

Let R be a ring with 1, and Q its maximal ring of right quotients. We give a number of necessary and sufficient conditions on R so that q is prime regular. We obtain as corollaries conditions so that Q is a full linear ring or Q is simple and satisfies xy = 1 implies yx = 1.

A subdirect decomposition of semiprime rings and its application to maximal quotient rings

Levy [2] has examined semiprime rings which are irredundant subdirect products of prime rings. In this note we look at the role of inessential prime ideals and see how every semiprime ring is a subdirect product of (i) a semiprime ring which is an irredundant subdirect product of prime rings, and (ii) a semiprime (nonprime) ring, all of whose prime ideals are essential. This leads to a direct sum decomposition of maximal left quotient rings of semiprime rings with left singular ideal zero.

Classical quotient rings

Throughout R is a ring with right singular ideal Z ( R ) Z(R) . A right ideal K of R is rationally closed if x − 1 K = { y ∈ R : x y ∈ K } {x^{ - 1}}K = \{ y \in R:xy \in K\} is not a dense right ideal for all x ∈ R − K x \in R - K . A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R / Z ( R ) R/Z(R) is semiprime, Z ( R ) Z(R) is rationally closed, and Z ( R ) Z(R) contains no closed uniform right ideals. These rings include the quasi-Frobenius rings as well as the semiprime Goldie rings. The commutative Cl-rings have Cl-classical quotient rings. The injective ones are congenerator rings. In what follows, R is a Cl-ring. A dense right ideal of R contains a right nonzero divisor. If R satisfies the minimum condition on rationally closed right ideals then R has a classical Artinian quotient ring. The complete right quotient ring Q (also called the Johnson-Utumi maximal quotient ring) of R is a Cl-ring. If R has the additional property that bR is dense whenever b is a right nonzero divisor, then Q is classical. If Q is injective, then Q is classical.

Regular self-injective rings with a polynomial identity

This paper studies maximal quotient rings of semiprime P. I.-rings; such rings are regular, self-injective and satisfy a polynomial identity. We show that the center of a regular self-injective ring is regular self-injective; this enables us to establish that the center of the maximal quotient ring of a semiprime P. I.-ring R is the maximal quotient ring of the center of R, as well as some other relationships. We give two decompositions of a regular self-injective ring with a polynomial identity which enable us to show that such rings are biregular and are finitely generated projective modules over their center.

Prime Ideals in Regular Self-Injective Rings

Although the notion of the maximal quotient ring of a nonsingular ring has been around for some time, not much is known about its structure in general beyond the important theorems of Johnson and Utumi [4; 11] that it is von Neumann regular and self-injective. The purpose of this paper is to study the structure of such a regular, self-injective ring R by looking at its prime ideals. Initially, we show that the primes of R separate into two types, called ‘'essential” and ‘“closed”, and that for any prime P, the two-sided ideals in the ring R/P are linearly ordered.

Quotient Rings of a Class of Lattice-Ordered-Rings

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R(i) (d1q)+ Λ (d2q)- = 0, and(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.We remark that this characterization holds even when R does not have an identity element.

Vascular Adjustments in Dilatory Reactions

Dilatory responses of the isolated intestinal vasculature of the rat have been studied during constant infusion of acetylcholine (Ach), isoproterenol (IP) andmyotropic, i.e. Ca<sup>++</sup> antagonistically acting doses of propranolol. The initial vasodilation was followed by a succeeding increase of flow resistance. This effect, called <i>‘escape phenomenon’ </i>could constantly be proved during application of Ach, was dosedependent and reached a maximal escape quotient of nearly 30%. Large vasoconstrictive doses of Ach also showed an escape phenomenon indicating the probability of one basal mechanism causing the escape from vasodilation and vasoconstriction. Vasodilations evoked by infusion of IP and large doses of β-blockers showed escape reactions only in some cases. The discrepancy between the different escape behaviour can be understood by comparing the rate of development of the initial vasodilation, which was two times slower for IP and propranolol than for Ach. The escape from Ach-elicited vasodilations could be clearly reduced by myotropic acting doses of LB 46. These results demonstrate that the escape from vasodilation is not a drug-specific phenomenon but is of <i>myogenic </i>origin. The analogy of the escape from vasodilation and vasoconstriction suggests a common basal myogenic mechanism perhaps due to changes of Ca<sup>++</sup>-fluxes. Thus, the initial vascular reaction would represent an ‘overshoot’ and the escape an adjustment reaction in a ‘steady-state system’.

Maximal Quotient Rings and S-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.

The Singular Congruence and the Maximal Quotient Semigroup

It is a well known result (see [4, p. 108]) that if R is a ring and Q(R) its maximal right quotient ring, then Q(R) is (von Neumann) regular if and only if every large right ideal of R is dense. This condition is equivalent to saying that the singular ideal of R is zero. In this note we show that the condition loses its magic in the theory of semigroups.

Evaluation of left ventricular function from isovolumic pressure measurements during isometric exercise

In 24 patients left ventricular (tipmanometer) and aortic pressure measurements were carried out before and during a standardized handgrip. On the basis of findings at diagnostic cardiac catheterization, the patients were divided into 2 groups. Group 1 consisted of 13 patients without or with minimal loading of the left ventricle; group 2 comprised 11 patients with a primarily pressure-loaded left ventricle or coronary heart disease. During handgrip left ventricular systolic pressure, end-diastolic pressure and heart rate increased significantly in both groups. However, the extent of the increase of left ventricular end-diastolic pressure was considerably greater in group 2 (+8.2 mm Hg) than in group 1 (+2.0 mm Hg). Both left ventricular maximal dP/dt and the maximal quotient (dP/dt)/P, which represents a measure of the velocity of shortening of the contractile elements, increased significantly in group 1; in group 2 only maximal dP/dt increased significantly, and this increase was only about half of that observed in group 1. The time intervals from the onset of contraction to the points of maximal dP/dt and maximal (dP/dt)/P, respectively, shortened significantly during handgrip in group 1 and remained unchanged in group 2. We conclude that (1) left ventricular contractile function is impaired in patients in group 2 compared with that in group 1; (2) the primary response of the normal or near normal left ventricular myocardium to an acute load produced by handgrip is enhancement of the contractile state; and (3) the compromised left ventricular myocardium is most likely to utilize the Frank-Starling mechanism to generate increased pressure.