SKEW POLYNOMIAL RINGS WHICH ARE GENERALIZED ASANO PRIME RINGS

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.

In this paper we shall give a new proof of the well-known theorem of Faith-Utumi[1]. Using our method we can show that every right order ofK n is a prime right Goldie ring, whereK n is the n×n-matrix ring over division ring K. Specially,D n is a prime right Goldie ring, ifD is a right order ofK.

Let R be a prime ring and δ a derivation of R. Divided powers \( D_n ^{\underline{\underline {def.}} } \tfrac{1} {{n!}}\tfrac{{d^n }} {{dx^n }} \) of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that \( S^{\underline{\underline {def.}} } Q[x;\delta ] \) is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form $$ \theta (f) = \sum\limits_{n = 0}^\infty {\zeta _n D_n (f)} forf \in Q[x;\delta ], $$ where ζ n ∈ C S (R), the centralizer of R in S. As an application, we also use the Ore extension R[x; δ] to deduce Kharchenko’s theorem for a single derivation. These results are extended to the Ore extension R[X;D] of R by a sequence D of derivations of R.

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].

Let $R = \\oplus_{n \\in \\Z} R_{n}$ be a strongly graded ring of type $\\Z$ and $R_{0}$ is a prime Goldie ring. It is shown that the following three conditions are equivalent: (i) $R_{0}$ is a $\\Z$-invariant Krull ring, (ii) $R$ is a Krull ring and (iii) $R$ is a graded Krull ring. We completely describe all $v$-invertible $R$-ideals in $Q$, where $Q$ is a quotient ring of $R$. 

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

In this paper we study right Mori Orders, which are those prime Goldie rings that satisfy the ascending chain condition on integral right ν-ideals. We examine the right Mori property of two local orders with identical prime ideals, and show that if one of them is a right Mori order so is the other. As one of the main results, it is proved that the intersection of a family of right Mori orders of finite character is again a right Mori order. This implies that non-commutative Krull orders in the sense of Marubayashi are left as well as right Mori orders.

On generalized Dedekind prime rings

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.

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.