Rings whose faithful left ideals are cofaithful

The polynomial ring over a Goldie ring need not be a Goldie ring

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.

Let R be a ring and S a strictly totally ordered monoid. Let ω: S → End(R) be a monoid homomorphism. Let M R be an ω-compatible module and either R satisfies the ascending chain conditions (ACC) on left annihilator ideals or every S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join. We show that M R is p.q.-Baer if and only if the generalized power series module M[[S]] R[[S, ω]] is p.q.-Baer. As a consequence, we deduce that for an ω-compatible ring R, the skew generalized power series ring R[[S, ω]] is right p.q.-Baer if and only if R is right p.q.-Baer and either R satisfies the ACC on left annihilator ideals or any S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join in R. Examples to illustrate and delimit the theory are provided.

We are mainly interested in the study of prime and semi-prime rings and their rings of quotients. However, our argument proceeds largely in the category of modules (§ 1 to 4) and bimodules (§ 5 to 7).After a brief description of the generalized rings of quotients introduced recently by Johnson, Utumi, and Findlay and the present author, we study a closure operation on the lattice of submodules of a module. For the lattice of left ideals of a ring, the concept of closed submodules reduces to the If-ideals of Utumi. The lattice of closed submodules of a module is always a complete modular lattice. We are specially interested in the case when it is a complemented lattice. This happens, in particular, when the singular submodule of Johnson and Wong vanishes. We consider the lattice of closed right ideals of a prime ring S and determine the maximal ring of right quotients of S in the case when this lattice has atoms.

This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation— QF- 3 3 algebras—in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: R R contains a faithful Σ \Sigma -injective left ideal, R R contains a faithful I I II -projective injective left ideal; the injective hulls of projective left R R -modules are projective, and the projective covers of injective left R R -modules are injective. Moreover, these rings are shown to be semi-primary and to include all left perfect rings with faithful injective left and right ideals. The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary. In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring.

We define a ring R to be an FL (full linear) ring if R is isomorphic to the full ring of linear transformations of a left vector space over a division ring.R is QFL if its left maximal quotient ring is an FL ring.In this paper we give necessary and sufficient conditions for a ring to be a QFL ring.We also generalize some results of Chase and Faith concerning subdirect sum decompositions of rings whose left maximal quotient ring is the direct product of FL rings.If rM is a left i?-module, then C(rM) will denote the set of closed submodules of M. We say that C(rM) is atomic if C(rM) contains minimal nonzero elements, called atoms, and each nonzero element of C(rM) contains at least one atom.rM is Q-prime if for any two atoms Ii and h o C(rM), there exists isomorphic submodules Ai and X2 of M such that A, is essential in Ii for i = 1 and 2. If A is a subset of M, then Ir(X) will denote the left annihilator in R of X. Theorem\. R is QFL if and only if the following conditions are satisfied : (i) Z(RR)=0, (ii) C(rR) is atomic, (iii) rR is Q-prime.Proof.Let Q be the left maximal quotient ring of R. Ii conditions (i)-(iii) hold, then, by [2, p. 70], C(0(?) = C(i?) under contraction, and so C(qQ) is atomic.Since Z(rR) = 0, we have that Q is a left selfinjective regular ring.Thus A is an atom of C(qQ) if and only if A is a minimal left ideal of Q.For the sake of simplicity of notation we will refer to the atoms of C(qQ) and C(rR) as atoms of Q and R respectively.Suppose Ax and A2 are atoms of Q.By [2, p. 70], Ax and A2 are injective as left i?-modules.If Ix=Axf~\R and I2 = AiC\R, then lx and Ii are atoms of R. Since rR is Q-prime, there exists isomorphic left ideals Xi and A2 of R such that X, is essential in i", for i= 1 and 2. Clearly, Xi is essential in Ai for i= 1 and 2. The injectivity of Ai gives that Ax and Ai are isomorphic.Thus Ir(Ax) =Ir(A).

Emerging trends in soft set theory and related topics.

Let a be an idempotent kernel functor defining the ring of left quotients Q"(R).We introduce a notion of cr-divisibility, and show that a cr-torsionfree /{-module M is a module over Qa(R) if and only if M is cr-divisible.In his paper Rings and modules of quotients, Goldman [2] investigated the construction of rings of quotients of a ring R with respect to idempotent kernel functors of the category RM of left -modules.If a is such an idempotent kernel functor, then it was shown that every cr-torsionfree and o--injective left -module is a left module over the ring of quotients Qa(R).We have called a module RM o--divisible if it is generated in the categorical sense by the A-module Qa(R).In Theorem 4 below we show that a atorsionfree module RM is a left ff(F)-module (extending the action of R) if and only if M is cr-divisible.In Theorem 6 we show that the quotient functor Q" is naturally isomorphic to the functor Qa(R)_if and only if every ff-torsionfree and cr-divisible left ^-module is cr-injective.All rings considered in this note are associative rings with identity, and all modules are assumed to be unital.For the convenience of the reader we repeat the relevant definitions of [2] and [3].A functor o from RM into itself is called a kernel functor if it has the following properties: (i) a(M) is a submodule of M for every M e RM. (ii) Iff: M->-N is a homomorphism, then f(o(M))^ o(N), and o(f) is the restriction off to o(M).(iii) If M' is a submodule of M-, then a(M') = M' t~\a(M).If, in addition, o(Mo(M))=0for all M e RM, then a is called an idempotent kernel functor.For the remainder of the note we let a be a fixed idempotent kernel functor of RM.A module RM is called a-torsion if o(M) = M; it is called cr-torsionfree if o(M)=0.A submodule M' of M is said to be cr-dense in M if M\M' is cr-torsion ; it is said to be tr-closed in M if M\M' is cr-torsionfree.The intersection of all cr-closed submodules of M which contain M' is called the o--closure of M' in M.

Let R be a semisimple Artin ring (i.e. R is a direct sum of matrix rings over division rings). Then if L is a left ideal of R, L = Re where e is an idempotent and so L is exactly the left annihilator of the element 1 -e. We investigate the structure of rings having this property. If S is a subset of a ring R, let e(S) and r(S) denote the left and right annihilators of S. The notation eR(S) will be used when it is necessary to specify the ring R. DEFINITION. R is a left elemental annihilator ring (l.e.a.r.) if, whenever L is a left ideal of R, there exists an element aER such that L=e(a). A right elemental annihilator ring (r.e.a.r.) is defined analogously. Notice that if R is any ring (always assumed to have a unity element) and S is a subset of R, then e(S)=e(r(e(S))) and r(S) =r(e(r(S))). Hence if R is a l.e.a.r., and L is a left ideal, we have in particular that L==e(r(L)). If R is a r.e.a.r., and I a right ideal, then I=r(e(I)). We consider first the commutative case. It is known [1, Theorem 1.1] that if R is completely primary (local with nilpotent maximal ideal), then R has the property that every ideal is the annihilator of some subset of R if and only if R has a unique minimal ideal. Using the methods of Theorem I below, it is possible to derive from this the fact that if R is a commutative noetherian ring, then every ideal of R is the annihilator of some subset if and only if R is a direct sum of completely primary rings each of which has a unique minimal ideal. By imposing the more strenuous condition that R actually be an elemental annihilator ring, we obtain a similar result without the hypothesis of the chain condition.

All rings are assumed to be associative and with nonzero identity element, and all modules are assumed to be unitary. A ring is said to be right (resp., left) invariant if all right (resp., left) ideals of it are ideals. Expressions such as “an invariant ring” mean that the corresponding right and left conditions hold. A right module M over a ring A is called a multiplication module if for every submodule N of M , there exists an ideal B of the ring A such that N = MB. A module MA is said to be flat if for every left A-module L, the natural group homomorphism M ⊗A L is a monomorphism. If all submodules of any right or left module over a ring A are flat, then one says that the weak global dimension of the ring A does not exceed 1; in this case, we write w.gl.dim(A) ≤ 1 for brevity. In [7], it is proved that every faithful multiplication module over a commutative ring is a flat module. If A is a commutative ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive semiprime ring [5]. If A is an invariant semiprime ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive ring [13, Theorem 7.31]. The main result of the present paper is Theorem 1.

We study classical right quotient rings and ordinary extensions of various kinds of 2-primal rings, constructing examples for situations that raise naturally in the process. We show: (1) Let R be a right Ore ring with P(R) left T-nilpotent. Then Q is a 2-primal local ring with P(Q)=J(Q) = {ab-1 ∈ Q | a ∈ P(R), b ∈ C(0)} if and only if C(0)=C(P(R))=R∖P(R), where Q is the classical right quotient ring of R. (2) Let R be a right Ore ring. Then R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R). As a consequence, if R is a right Noetherian ring, then R[[x]] is a domain whose classical right quotient ring is a division ring if and only if R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R).

Throughout this note, R stands for a ring with identity, and all modules are unital R-modules. In this note, we ask when the class of all finitely generated reflexive (torsionless) left modules is closed under extensions. This is not always the case, even if R is left and right noetherian (cf. Bass [2, Lemma 3.2] and [3, p. 12]). As will be shown, in case R is a commutat ive noetherian local domain of Krull dimension two, R is a CohenMacaulay ring whenever the class of the finitely generated reflexive modules is closed under extensions. Our main aim is to show that if R is a left and right noetherian ring then both the class of all finitely generated torsionless left modules and the class of all finitely generated reflexive left modules are closed under extensions if and only if for i = 1, 2 the functor Ext , 1 (Ext~ ( , R), R) vanishes on the finitely generated right modules. Note that the latter condition is satisfied if R is a commutat ive Gorenstein ring (see Bass [3]). We will show also that if R is a left and right noetherian ring with a minimal injective resolution 0 ~ RR ~ Eo ~ E1 , . . . such that weak dim E l < i + 1 for i = 0, 1 then both the class of finitely generated torsionless left modules and the class of finitely generated reflexive left modules are closed under extensions. In the proof, it will be shown that if R is a left and right noetherian ring with inj dim RR _--< 2 then the class of the finitely generated reflexive left modules is closed under extensions. On the other hand, even if R is a left and right noetherian ring with inj dim RR = inj dim R R < 2, the class of all finitely generated torsionless left modules is not necessarily closed under extensions. In what follows, we denote by ( )* both the R-dual functors, and for a given module M we denote by eu: M ~ M** the usual evaluation map and by E(M) the injective envelope of M. Recall that a module M is said to be torsionless if eu is a monomorph i sm and to be reflexive if eu is an isomorphism. Note that a module M is torsionless if and only if it is cogenerated by R. We denote by mod R (resp. mod R ~ the category of the finitely presented left (resp. right) modules. Note that if R is left noetherian then every finitely generated left module is finitely presented.

In this paper, we show how to reduce the study of nondegenerate local Goldie (-1, 1) rings to the strongly prime case, via the notions of uniform ideals and essential subdirect product. Also, we construct the maximal left quotient ring of (-1, 1) ring that is a left quotient ring of itself. We follow Utumi where a maximal left quotient ring is constructed as a direct limit of a partially defined homomorphism from the left ideal of R to R.

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 $R \geq 1$.

This paper investigates two related problems. The first is to describe the double centralizer of an arbitrary projective right iϋ-modiile. This proves to be the ring of left quotients of R with respect to a certain canonical hereditary torsion class of left ϋί-modules determined by the projective module. The second is to determine the relationship between rings of left quotients of R and S, where S is the endomorphism ring of a finitely generated projective right ϋί-module PR. It is shown that there exists an inclusion-preserving, one-to-one correspondence between hereditary torsion classes (or localizing subcategories) of left ^-modules and hereditary torsion classes of left E-modules which contain the canonical torsion class determined by PR. If QR and Qs are rings of left quotients with respect to corresponding classes, then P($$RQR is a finitely generated projective right Q^-module with Qs as its Q#-endomorphism ring. Necessary and sufficient conditions are obtained for the maximal rings of left quotients to be related in this manner. In particular, this occurs when PR is a faithful .R-module and R is either a semi-prime ring or a ring with zero left singular ideal. The situation considered includes the case where S is an arbitrary ring, SP is a left ^-generator, and R is the Sendomorphism ring of SP When SP is a projective left Sgenerator, the maximal rings of left quotients of R and S are related in the manner considered above.