Centred bimodules over prime rings: Closed submodules and applications to ring extensions

Perfect Gabriel filters of right ideals and their corresponding right rings of quotients have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of quotients have a symmetry that mimics the commutative case. In this paper, we study rings of quotients that combine these two desirable properties. We define the symmetric versions of a right perfect ring of quotients and a right perfect Gabriel filter — the perfect symmetric ring of quotients and the perfect symmetric Gabriel filter and study their properties. Then we prove that the standard construction of the total right ring of quotients [Formula: see text] can be adapted to the construction of the largest perfect symmetric ring of quotients — the total symmetric ring of quotients [Formula: see text]. We also demonstrate that Morita's construction of [Formula: see text] can be adapted to the construction of [Formula: see text].

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.

Let K be a field and K[x] the polynomial ring over K in an indeterminate x. If P is a prime ideal of K[x], then there exists an irreducible polynomial f in K[x] such that P = K[x]f. This result is quite old and basic; however no corresponding result seems to be known for a polynomial ring in n indeterminates x1, ..., xn over K. Actually, it seems to be very difficult to find some system of generators for a prime ideal of K[xl,...,Xn]. Now, K [x1, ..., xn] is a Noetherian ring and by a converse of the principal ideal theorem for every prime ideal P of K[x1, ..., xn] there exist n polynomials fi, f..n such that P is minimal over (fi, ..., fn), the ideal generated by { fl, ..., f}n ([4], Theorem 153). Also, as a consequence of ([1], Theorem 1) it follows that any prime ideal of K[xl, X.., Xn] is determined by n polynomials. However it is not clear in [1] how to find these polynomials and no converse result is proved. The purpose of this paper is to show a result which in particular implies that every prime ideal of K[xi, ..., xn] is determined by a sequence of n polynomials which is in some sense irreducible, and such that the converse is also true. In general, let R be any prime ring. If P is a nonzero prime ideal of the polynomial ring R[x] with P n R = 0 (an R-disjoint prime ideal), then P = Q[x]fo n R[x] where Q is a ring of right quotients of R and fo E C[x] is an irreducible polynomial, C being the extended centroid of R ([2], Corollary 2.7). This characterization has the problem that we have to compute Q to have a prime ideal determined. A better way (an intrinsic one) to determine an R-disjoint prime ideal has been given in [3]. We proved that every R-disjoint prime ideal P of R[x] is determined by just one polynomial in P which is in some sense irreducible ([3], Theorem 1.4).

Let R be a prime ring with maximal right ring of quotients Q, extended centroid C and a noncommutative Lie ideal L. Our aim is to characterize the forms of generalized skew derivations $$\delta $$ and $$\varrho $$ of R in case of $$\begin{aligned} p(\delta (u)^{l_{1}}\varrho (u)^{l_{2}}\delta (u)^{l_{3}}\varrho (u)^{l_{4}}\cdots \varrho (u)^{l_{k}} )^{n}=0, \end{aligned}$$ for all $$u\in L$$ , where $$0 \ne p\in R$$ , $$l_{1},l_{2},\ldots ,l_{k}$$ are fixed nonnegative integers with $$l_{1}\ne 0$$ and n is a positive fixed integer. The proof of this main result is based on the so-called extended Jacobson density theorem, which is different from that of previous results in the literature. Also as a result, we obtain if $$p\delta (u)^{n}=0$$ for all $$u \in L$$ , then there exists $$a\in Q$$ such that $$\delta (x)=ax$$ for all $$x \in R$$ and $$pa=0$$ unless R satisfies $$s_4$$ .

Let R be a prime ring with extended centroid C, maximal right ring of quotients U, a nonzero ideal I and a generalized derivation δ. Suppose δ(x)n =(ax)n for all x ∈ I, where a ∈ U and n is a fixed positive integer. Then δ(x)=λax for some λ ∈ C. We also prove two generalized versions by replacing I with a nonzero left ideal [Formula: see text] and a noncommutative Lie ideal L, respectively.

LetA be a prime ring with maximal right ring of quotients Q and with extended centroid C. Further, let S be a set, let αS→ A be a map, let m be a positive integer and let Ei Fi S m-1, be maps. We study functional identities of the form for all s 1,s 2,⋯,sm ϵ S (where , etc). If Sα is an (m+ l)-free subset of Q (for example, if S α = A and A is not algebraic of bounded degree m+1 over C, or S α is a ncentral Lie ideal of A and A is not algebraic of bounded degree m + 2 over C), definitive results are obtained.

While the classical right ring of quotients Q cl r (R) exists (if and) only if R is a right Ore ring, a maximal right ring of quotients Qmaxr(R) exists for any ring R. To define Q max r (R), one must first understand the structure of the endomorphism ring of a QI (quasi-injective) module.

Let R be a prime Goldie ring with quotient ring Q and σ be an automorphism of R. We define (σ-) generalized Asano prime rings and prove that a skew polynomial ring R[x; σ] is a generalized Asano prime ring if and only if R is a σ-generalized Asano prime ring. This is done by giving explicitly the structure of all v-ideals of R[x; σ] in case R is a σ-Krull prime ring. We provide some examples of σ-generalized Asano prime rings which are not Krull prime rings.

Centers and prime ideals in group algebras of polycyclic-by-finite groups

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).

In [10], it is defined that a right (or left) ideal I of a ring R is very large if the cardinality of R/I is finite. It is also proven in [10, Theorem 3.4] that if R is a prime ring with 1 such that its characteristic is zero, then R is a right order in a simple ring with the minimum condition on one sided ideals if every large right ideal of R is very large. In the present note, we shall prove that if R is a prime ring with 1 such that its characteristic is zero and R is also a compact topological ring, then R is a right and left order in a simple ring with the minimum condition on one sided ideals, which is also a non-discrete locally compact topological ring if and only if every large right ideal of R is open. In particular, if R is an integral domain with 1 (not necessarily commutative) such that its characteristic is zero, then R is openly embeddable [13, p. 58] in a locally compact (topological) division ring if and only if every large right ideal of R is open. Following S. Warner [13], we shall say R is openly embeddable in a quotient ring Q(R) if there is a topology on Q(R) which is compatible with its structure, which induces on R its given topology and for which R is an open subset.

Let [Formula: see text] be a prime ring, [Formula: see text] a noncentral Lie ideal of [Formula: see text] and [Formula: see text] the extended centroid of [Formula: see text], where [Formula: see text] is the Utumi quotient ring of [Formula: see text]. Suppose that [Formula: see text] are two nonzero generalized derivations of [Formula: see text]. Recently, [Dhara et al., Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48(1) (2020) 154–167] studied the case [Formula: see text] for all [Formula: see text], where [Formula: see text] is a derivation (which is independent of [Formula: see text]) and [Formula: see text] fixed positive integers and obtained the structure of [Formula: see text] and the ring [Formula: see text]. [Liu, An Engel condition with two generalized derivations on Lie ideals of prime rings, J. Algebra and Its Applications, 22(5) (2023) 2350115] studied a similar identity replacing derivation [Formula: see text] with another generalized derivation [Formula: see text]. Recently, [Pandey, Pair of generalized derivations on Lie ideals in prime rings, Asian-European J. Math. 16(1) (2022) 2350002] studied an identity considering composition of mapping, that is [Formula: see text] for all [Formula: see text]. In this paper, we consider the situation containing composition of mappings as well as Engel condition, that is, [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed integers and then obtain the nature of mappings [Formula: see text], [Formula: see text] and the ring [Formula: see text].

Given arbitrary rings A, B such that A ⊂ B and an arbitrary prime ideal P of A, we show how to construct a quotient ring Ap such that A ⊂ Ap ⊂ B. The ring Ap contains a prime ideal P lying over P and the prime ring A/P may be embedded in Ap/P.

Torsion-freeness and non-singularity over right p.p.-rings

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.