Classical Quotient Ring Research Articles

Classical rings of quotients of semiprimePI-rings

Classical rings of quotients of semiprimePI-rings

Finite dimensional rings of quotients

In this paper we characterize commutative rings with finite dimensional classical ring of quotients. To illustrate the diversity of behavior of these rings we examine the case of local rings and FPF rings. Our results extend earlier work on rings with zero-dimensional rings of quotients.

Classical Quotient Rings of Group Graded Rings

Classical Quotient Rings of Group Graded Rings

Additive rank functions in noetherian rings

Let R be a ring with unit element. A function 1, that assigns to each finitely generated R-module M a nonnegative integer 1(M) such that J(M) = A(N) + 1(L) for each short exact sequence 0 + N--f M + L --f 0 of finitely right R-modules is called an additive rank function. Such a function is constant on isomorphism classes of finitely generated right R-modules, so that 1(M/cr(M)) = 0 whenever CI: A4 -+ M is an injective endomorphism of M. When R is semisimple artinian, then an additive rank function arises very naturally by defining A(M) = length(M), the number of simple direct summands in a decomposition of M. In 1964, Goldie [2] adapted this idea to finitely generated modules over a right noetherian ring, the resulting additive rank function is presently called the reduced rank. The concept lay dormant for over 10 years until Goldie started using it in 1976, providing new and strikingly elegant proofs of several important results of ring theory. Some of these appear in [3], others are included in [ 11. Since then, the reduced rank has become a widely used tool, particularly in the theory of noncommutative noetherian rings. We recall the definition of the reduced rank p(M) of a finitely generated right R-module M over a right noetherian ring R. If N denotes the nilpotent radical of R then R/N has a classical semisimple artinian quotient ring Q(R/N). For a finitely generated right R/N-module M set pRIN(M) = length(M@.,, Q(R/N)). Since R,NQ(R/N) is flat, pRIN is an additive rank function for finitely generated right R/N-modules. For a finitely generated right R-module M set

Completely 0-simple semigroups of quotients III

In a recent paper [6] the authors introduced the concept of a completely 0-simple semigroup of quotients. This definition has since been extended to the class of all semigroups giving a definition of semigroup of quotients which may be regarded as an analogue of the classical ring of quotients. WhenQis a sernigroup of quotients of a semigroupS, we also say thatSis an order inQ.

On functions connected with sequential absolute, cantor completion and classical ring of quotients

On functions connected with sequential absolute, cantor completion and classical ring of quotients

On zero-dimensional rings of quotients and the geometry of minimal primes

The purpose of this paper is to examine the ways in which the structure of the Minimal Spectrum (Min(A )) of a commutative ring A is reflected in its maximal flat mono-epimorphism (Q,,,(A)) and its classical ring of Quotients (Q,,(A)). In the reduced case, there is an extensive literature [Ml, H & J, Mew, Q] on this subject. We focus attention on the subtleties created by non-trivial nilpotents. In general, there are homomorphisms relating Q,,,(A) and Q,,(A) to T(Min(A), 8) (0 denotes the structure sheaf of A). Examples are given to show that even in global flat dimension 1, these three objects can be distinct. With this potential in mind, we ask how such pathology can be eliminated. In particular, we show that for any commutative ring A, the following are equivalent:

Semi-local localizations of rings and subdirect decompositions of modules

Let M denote a module category, let A and B be objects of M , and assume End M (A) possesses a classical right ring of quotients Q A . The goal of this paper is to describe the finitely generated projective right Q A modules in terms of certain subdirect decompositions (called marginal decompositions) of A n , for integers n. The idea is to abstract the category Q Ab of quasi-isomorphisms of torsion-free abelian groups to a noncommutative setting. We then investigate Azumaya-Krull-Schmidt uniqueness theorems for marginal decompositions of A. Applications include an exchange result and a classification of those A for which Q A is a semisimple ring.

Semi-perfect FPF rings and applications

A ring R is right FPF if each faithful finitely generated right R module is a generator of MOD- R. The purpose of this paper is to answer several questions posed by C. Faith and S. Page in their monograph (“FPF Ring Theory: Faithful modules and generators of MOD- R,” London Math. Soc. Lecture Note Series, Cambridge Univ. Press, 1984), and then to apply these results to semi-perfect left or right Noetherian right FPF rings. Thus, a semi-perfect ring R is right FPF iff (i) R possesses a semi-perfect, right self-injective classical (left and right) ring of quotients, (ii) faithful finitely generated right ideals of R are generators of MOD- R, and (iii) the basic ring of R is right strongly bounded. Furthermore, the set of left regular elements of R is the set of right regular elements of R is the set of elements regular modulo Z r ( R). The semi-perfect ring R is an injective cogenerator of MOD- R iff R is right FPF and right Kasch. Chain conditions on R are also considered.

The Row Rank of a Subring of a Matrix Ring

For $R$ a subring of an $n \times n$ matrix ring ${M_n}(\Delta )$ over a division ring $\Delta$, we examine an invariant called the row rank of $R$. Roughly speaking, the row rank of $R$ is the largest integer $k$ such that $R$ contains all $k$-rowed matrices over a left order in $\Delta$. The row rank of $R$ is then an integer between 0 and $n$; and we will see that row rank $R \geq 1$ means that ${M_n}(\Delta )$ is the maximal left quotient ring of $R$, while row rank $R = n$ signifies that ${M_n}(\Delta )$ is the classical left quotient ring of $R$. Thus row rank provides a link between maximal and classical quotient rings for rings of this type. A description of the subrings $R$ with row rank $R \geq k$ is obtained which subsumes and generalizes earlier theorems of Faith-Utumi and Zelmanowitz, respectively, for the cases row rank $R = n$ and row rank $R \geq 1$.

The row rank of a subring of a matrix ring

For R R a subring of an n × n n \times n matrix ring M n ( Δ ) {M_n}(\Delta ) over a division ring Δ \Delta , we examine an invariant called the row rank of R R . Roughly speaking, the row rank of R R is the largest integer k k such that R R contains all k k -rowed matrices over a left order in Δ \Delta . The row rank of R R is then an integer between 0 and n n ; and we will see that row rank R ≥ 1 R \geq 1 means that M n ( Δ ) {M_n}(\Delta ) is the maximal left quotient ring of R R , while row rank R = n R = n signifies that M n ( Δ ) {M_n}(\Delta ) is the classical left quotient ring of R R . Thus row rank provides a link between maximal and classical quotient rings for rings of this type. A description of the subrings R R with row rank R ≥ k R \geq k is obtained which subsumes and generalizes earlier theorems of Faith-Utumi and Zelmanowitz, respectively, for the cases row rank R = n R = n and row rank R ≥ 1 R \geq 1 .

Clifford semigroups of left quotients

Several definitions of a semigroup of quotients have been proposed and studied by a number of authors. For a survey, the reader may consult Weinert's paper [8]. The motivation for many of these concepts comes from ring theory and the various notions of rings of quotients. We are concerned in this paper with an analogue of the classical ring of quotients, introduced by Fountain and Petrich in [3].

Classical quotient rings of trivial extensions

Classical quotient rings of trivial extensions

Brandt semigroups of quotients

In an earlier paper [3], we introduced a new notion of completely 0-simple semigroup of quotients. The definition is motivated by a connection between Artinian simple rings and completely 0-simple semigroups. Given an Artinian simple ring Q, one obtains a completely 0-simple multiplicative sub-semigroup by taking the union of the subsets eiQej(i, j = l,…, n), where 1 = e1 + … + en and the ei are primitive orthogonal idempotents. If Q is the classical ring of left quotients of R, then is a semigroup of left quotients of in the sense that every element q in can be written as q = a-1b for some elements a, b in with a2 ≠ 0.

Semicritical Rings and the Quotient Problem

AbstractThe purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

On noncommutative localization

For a prime ideal P of a Noetherian ring R, Goldic defined in 17, 81 a localization Q of R at P. This localization is constructed as a subring of the inverse limit of the rings Q,,(R/P’“‘), where P(“) is the nth symbolic power of P. Note that R/P’“’ has an Artinian classical ring of quotients, which is denoted by Q,.,(R/P’“‘). The ring Q has been shown by Goldie to have the property that Q/J(Q) z Q,,(R/P) and fl?=, J(Q)’ = (0), where J(Q) is the Jacobson radical of Q. The inversive localization Rr,,, of R at P, introduced by Cohn in (61, is defined as follows. If r(P) denotes the set of matrices regular modulo P, then R T(,,j is the ring universal with respect to the property that all matrices in r(P) are invertible over R,.o,. It has been shown in [ 11 that RrcJ,, is universal with respect to the property that R I ,,,/J(R,.,,,) is naturally isomorphic to Q,,(R/P), and so there is a canonical mapping from R,-(,,, into Goldie’s localization Q. Theorem 2.4 shows that Q z R,,,,/T, where T is the intersection of powers of J(Rrc,,). Theorem 2.6 then characterizes Q as the ring universal with respect to the property that Q/J(Q) is naturally isomorphic to Q,,(R/P) and nz-, J(Q)” = (0). The results relating the two localizations depend on Theorem 1.4, which shows that if R is a left Noetherian ring with prime radical N, then R,(,,, is a left Artinian ring. In fact, R,.,,v, z Q,,(R/K), where K is the smallest ideal of R such that every element regular modulo N is also regular modulo K. This implies that R,(,, is left Artinian when P is a minimal prime ideal, and makes it possible to define the symbolic powers of any semiprime ideal. Comments at the end of the paper indicate how these results can be extended to rings with finite reduced rank (in the sense of 121). and thus the results hold for all rings with Krull dimension.

Marubayashi-Krull orders and strongly graded rings

1. INTRODUCTION The aim of this note is to give sufficient conditions for a strongly Z-graded ring to be a Marubayashi-Krull order (cf. Theorem 3.10). Concrete examples of strongly E-graded rings are skew group rings over Z and generalized Rees rings. We also prove that under the same conditions the positive part of a strongly Z-graded ring is a Marubayashi-Krull order (cf. Theorem 4.1). Finally, we examine when these conditions may be weakened. 2. PRELIMINARIES A ring R is said to be a strongly B-graded ring if there exists a family of additive subgroups R, (n E Z) of R such that R = OnpLRn and wm =Rn+m for all n, m E Z. The elements of h(R) = lJneh R, are called the homogeneous elements of R. If x E R,, then x is said to be homogeneous of degree n. If x = C x, (x, E R,), then the elements x, are called the homogeneous components of x. Let I be an ideal of R. Denote by Ig the ideal generated by the homogeneous elements contained in I. I is said to be graded if I = Ig. In this case Z= RI, =Z,R, where I, = Zn R,. Details about strongly graded rings may be found in [lo]. If I is an ideal of R, then C(I) denotes the set of elements regular modulo I. If C(I) is a left and right Ore set in R, then Rcc,, denotes the left and right localization of R towards this Ore set C(1). If R is a prime Goldie ring, then Q(R), the classical ring of quotients of R, is a simple Artinian ring. The Asano overring of R is defined as S(R) = {x E Q(R) 1 Ix c R for some nonzero ideal I of R}. A prime Goldie ring R is said to be a Marubayashi-Krull order (or for short an M-Krull order) if:

Rings with Enough Invertible Ideals

All rings are associative with identity element 1 and all modules are unital. A ring has enough invertible ideals if every ideal containing a regular element contains an invertible ideal. Lenagan [8, Theorem 3.3] has shown that right bounded hereditary Noetherian prime rings have enough invertible ideals. The proof is quite ingenious and involves the theory of cycles developed by Eisenbud and Robson in [5] and a theorem which shows that any ring S such that R ⊆ S ⊆ Q satisfies the right restricted minimum condition, where Q is the classical quotient ring of R. In Section 1 we give an elementary proof of Lenagan's theorem based on another result of Eisenbud and Robson, namely every ideal of a hereditary Noetherian prime ring can be expressed as the product of an invertible ideal and an eventually idempotent ideal (see [5, Theorem 4.2]). We also take the opportunity to weaken the conditions on the ring R.

Hereditary Finitely Generated Algebras Satisfying a Polynomial Identity

If $\Lambda$ is a right and left p.p. ring which satisfies a polynomial identity and is a finitely generated algebra over its center, then $\Lambda \simeq \Gamma \times \Omega$, where $\Gamma$ is a semiprime ring having a von Neumann regular classical quotient ring which is module-finite over its center and $\Omega$ has nonzero prime radical at each of its Pierce stalks. If $\Lambda$ is right and left hereditary, then $\Gamma$ is an order over a commutative hereditary ring in the sense of [7]; the ring $\Omega$ is then a direct product of finitely many indecomposable piecewise domains.

The Quotient Problem for Noetherian Rings

Our work was motivated by attempts to find a criterion for the existence of a classical quotient ring, for a noetherian ring, in analogy with the various known criteria for the existence of an artinian classical quotient ring ([9], [10], [13], [2]).We have restricted our attention to Krull symmetric noetherian rings R, and we make heavy use of the fact that all their Krull composition factors are non-singular (Proposition 7). The collection Kprime R of the associated primes of the Krull composition factors of R plays a central role, taking the place of the collection of the associated primes of R.