In this paper, we introduce and study a strict generalization of symmetric rings. We call a ring $R \,\,\, 'P-symmetric'$ if for any $a,\, b,\, c\in R,\, abc=0$ implies $bac\in P(R)$, where $P(R)$ is the prime radical of $R$. It is shown that the class of $P$-symmetric rings lies between the class of central symmetric rings and generalized weakly symmetric rings. Relations are provided between $P$-symmetric rings and some other known classes of rings. From an arbitrary $P$-symmetric ring, we produce many families of $P$-symmetric rings.
With the notion of prime submodule defined by F. Raggi et.al. we prove that the intersection of all prime submodules of a Goldie module $M$, is a nilpotent submodule provided that $M$ is retractable and $M^{(Λ)}$-projective for every index set $Λ$. This extends the well known fact that in a left Goldie ring, the prime radical is nilpotent.
Prime and Semiprime Ideals in Commutative Ternary $Γ$-Semirings: Quotients, Radicals, Spectrum
Government College (A), Rajahmundry, A.P., India（拉贾姆恩德里政府学院(A)）；Acharya Nagarjuna University, Guntur, A.P., India（阿查里亚纳加琼纳大学）；Government College For Women(A), Guntur, Andhra Pradesh, India（贡图尔女子政府学院(A)）
Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, which coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-m-system sets, M-n-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "prime M-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this d…
The coincidence of the set of all nilpotent elements of a ring with its prime radical has a module analogue which occurs when the zero submodule satisfies the radical formula. A ring $R$ is 2-primal if the set of all nilpotent elements of $R$ coincides with its prime radical. This fact motivates our study in this paper, namely, to compare 2-primal submodules and submodules that satisfy the radical formula. A demonstration of the importance of 2-primal modules in bridging the gap between modules over commutative rings and modules over noncommutative rings is done and new examples of rings and m…
In this paper we continue our study of modules satisfying the prime radical condition ($\mathbb{P}$-radical modules), that was introduced in Part I (see \cite{BS}). Let $R$ be a commutative ring with identity. The purpose of this paper is to show that the theory of spectrum of $\mathbb{P}$-radical $R$-modules (with the Zariski topology) resembles to that of rings. First, we investigate the behavior of $\mathbb{P}$-radical modules under localization and direct sums. Finally, we describe the construction of a structure sheaf on the prime spectrum Spec$(M)$, which generalizes the classical struct…
Let R be a ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set B(R, G) of primes p such that p | |G| and R is not p-torsion free, is called the set of bad primes. When the ring is |G|-torsion free, i.e., B(R, G) is empty set, the properties of the rings R and R^G are closely connected. The aim of the paper is to show that this is also true when B(R, G) is not empty set under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (resp., the prime radical) of the ring R^G is equal to the intersection of the Jacobson radical (resp.…
A relationship between nilpotency and primeness in a module is investigated. Reduced modules are expressed as sums of prime modules. It is shown that presence of nilpotent module elements inhibits a module from possessing good structural properties. A general form is given of an example used in literature to distinguish: 1) completely prime modules from prime modules, 2) classical prime modules from classical completely prime modules, and 3) a module which satisfies the complete radical formula from one which is neither 2-primal nor satisfies the radical formula.
The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{\cal{P}M}:M)={\cal{P}}$" for every prime ideal ${\cal{P}}\supseteq {\rm Ann}(M)$, where $\sqrt[p]{\cal{P}M}$ is the intersection of all prime submodules of $M$ containing ${\cal{P}}M$). This class contains the family of primeful modules properly. This yields that over any ring all free modules and all finitely generated modules lie in the class of $\mathbb{P}$-radical modules. Also, we s…
The characterization of $H$-prime radical is given in many ways. Meantime, the relations between the radical of smash product $R # H$ and the $H$-radical of Hopf module algebra $R$ are obtained.
In this paper we introduce prime fuzzy ideals over a noncommutative ring. This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in [Not-so-fuzzy fuzzy ideals, Fuzzy Sets and Systems 37 (1990), 237--243], which lacks this equivalence in a noncommutative setting. Semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes. This allows us to introduce the fuzzy prime radical and contribute to establish the basis of a Fuzzy Noncommutative Ring Theory.
We introduce a complete radical formula for modules over non-commutative rings which is the equivalence of a radical formula in the setting of modules defined over commutative rings. This gives a general frame work through which known results about modules over commutative rings that satisfy the radical formula are retrieved. Examples and properties of modules that satisfy the complete radical formula are given. For instance, it is shown that a module that satisfies the complete radical formula is completely semiprime if and only if it is a subdirect product of completely prime modules. This g…
In this paper, we introduce the notion of pseudo-primary elements and pseudo-classical primary elements in an $L$-module $M$ and obtain their characterizations. The aim of the paper is to show $rad(N)\in M$, the radical of $N\in M$ is prime if $N$ is either pseudo-primary or pseudo-classical primary. Also, we study classical prime elements of an $L$-module $M$ to obtain many of its characterizations and its properties.
The Ritt problem asks if there is an algorithm that tells whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing if a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both prop…
Given a unital associative ring S and a subring R, we say that S is an ideal (or Dorroh) extension of R if for some ideal I of S, S = R + I, where the sum is direct. In this note we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These …
We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L+rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also co…
We answer several open questions and establish new results concerning differential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If $R$ is prime radical and $δ$ is a derivation of $R$, then the differential polynomial ring $R[X;δ]$ is locally nilpotent. This answers an open question posed in by Nielsen and Ziembowski. The nil radical of a differential polynomial ring $R[X;δ]$ takes the form $I[X;δ]$ for some ideal $I$ of $R$, provided that the base field is infinite. This answers an open question posed by Hong, Kim, Lee and Niels…
Let L be the Leavitt path algebra of an arbitrary directed graph E over a field K. This survey article describes how this highly non-commutative ring L shares a number of the characterizing properties of a Dedekind domain or a Prüfer domain expressed in terms of their ideal lattices. Special types of ideals such as the prime, the primary, the irreducible and the radical ideals of L are described by means of the graphical properties of E. The existence and the uniqueness of the factorization of a non-zero ideal of L as an irredundant product of prime or primary or irreducible ideals is establis…
In the set of continuous functions C(X,Y) where Y has a topology close to being discrete, there is an equivalence relation on X which characterizes the quasi-components of X. If Y satisfies weak algebraic conditions with a single binary operation then a stable set of functions forms an object generalizing an ideal of a ring. Calling such sets ideals there is a concept of a prime ideal. The ideal of functions vanishing on a quasi-component are prime ideals of C(X,Y). If Y has a zero set that is open then these prime ideals are min- max implying that when Y is a ring all of the prime ideals of C…
