Md Fazlul Hoque, Akhil Chandra Paul
Let $U$ be a Lie ideal of a 2-torsion free prime $Γ$-ring $M$ such that $uαu\in U$ for all $u\in U$ and $α\inΓ$. If $T:M\rightarrow M$ is an additive mapping satifying the relation $T(uαu)=T(u)αu$ for all $u\in U$ and $α\inΓ$, then we prove that $T(uαv)=T(u)αv$ for all $u, v\in U$ and $α\inΓ$. Also this result is extended to semiprime $Γ$-rings.
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In this article, we prove Conjecture $1$ posed in 2013 by Fo$\check{s}$ner \cite{fos} and Conjecture $1$ posed in 2014 by Ali and Fo$\check{s}$ner \cite{ali} related to generalized $(m,n)$-Jordan centralizer and derivation respectively.
In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fošner, \textit{On Generalized $(m, n)$-Derivations and Generalized $(m, n)$-Jordan Derivations in Rings,} Algebra Colloq. \textbf{21} (2014), 411--420] and [A. Fošner, \textit{A note on generalized $(m,n)$-Jordan centralizers,} Demonstratio Math. \textbf{46} (2013), 254--262]. Precisely, when $R$ is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized $(m,n)$-Jordan derivati…
Let $M$ be a noncommutative 2-torsion free semiprime $Γ$-ring satisfying a certain assumption and let $S$ and $T$ be left centralizers on $M$. We prove the following results: \\(i) If $[S(x),T(x)]_{α}βS(x)+S(x)β[S(x),T(x)]_{α}$=$0$ holds for all $x\in M$ and $α,β\in Γ$, then $[S(x),T(x)]_{α}$=$0$. \\(ii) If $S\neq 0 (T\neq 0)$, then there exists $λ\in C$,(the extended centroid of $M$) such that $T$=$λαS(S=λαT)$ for all $α\in Γ$. \\(iii) Suppose that $[[S(x),T(x)]_{α},S(x)]_{β}$=$0$ holds for all $x\in M$ and $α,β\inΓ$. Then $[S(x),T(x)]_{α}$=$0$ for all $x\in M$ and $α\inΓ$. \\(iv) If $M$ is a…
Semiprime ideals of an arbitrary Leavitt path algebra L are described in terms of their generators. This description is then used to show that the semiprime ideals form a complete sublattice of the lattice of ideals of L, and they enjoy a certain gap property identified by Kaplansky in prime spectra of commutative rings. It is also shown that the totally ordered sets that can be realized as prime spectra of Leavitt path algebras are precisely those that have greatest lower bounds on subchains and enjoy the aforementioned gap property. Finally, it is shown that a conjecture of Kaplansky regardi…
We study prime and semiprime Novikov algebras. We prove that prime nonassociative Novikov algebra has zero nucleus and center. It is well known that an ideal of an alternative (semi)prime algebra is (semi)prime algebra. We proved this statement for Novikov algebras. It implies that a Baer radical exists in a class of Novikov algebras. Also, we proved that a minimal ideal of Novikov algebra is either trivial, or a simple algebra.
Let R be a ring with derivation d, such that (d(xy))^n =(d(x))^n(d(y))^n for all x,y in R and n>1 is a fixed integer. In this paper, we show that if R is a prime, then d = 0 or R is a commutative. If R is a semiprime, then d maps R in to its center. Moreover, in semiprime case let A = O(R) be the orthogonal completion of R and B = B(C) be the Boolian ring of C, where C is the extended centroid of R, then there exists an idempotent e in B such that eA is commutative ring and d induce a zero derivation on (1-e)A.
Let $R$ be a commutative ring with $1$ and $n$ a natural number. We say that a submodule $N$ of $R^n$ is semiprime if for every $f=(f_1,\ldots,f_n) \in R^n$ such that $f_i f \in N$ for $i=1,\ldots,n$ we have $f \in N$. Our main result is that every semiprime submodule of $R^n$ is equal to the intersection of all prime submodules containing it. It follows that every semiprime left ideal of $M_n(R)$ is equal to the intersection of all prime left ideals that contain it. For $R=k[x_1,\ldots,x_d]$ where $k$ is an algebraically closed field we can rephrase this result as a Nullstellensatz for $M_n(R…
In this paper, we study on the primeness and semiprimeness of a Morita context related to the rings and modules. Necessary and sufficient conditions are investigated for an ideal of a Morita context to be a prime ideal and a semiprime ideal. In particular, we determine the conditions under which a Morita context is prime and semiprime.
For an $R$-module $M$, projective in $σ[M]$ and satisfying ascending chain condition (ACC) on left annihilators, we introduce the concept of Goldie module. We also use the concept of semiprime module defined by Raggi et. al. in \cite{S} to give necessary and sufficient conditions for an $R$-module $M$, to be a semiprime Goldie module. This theorem is a generalization of Goldie's theorem for semiprime left Goldie rings. Moreover, we prove that $M$ is a semiprime (prime) Goldie module if and only if the ring $S=End_R(M)$ is a semiprime (prime) right Goldie ring. Also, we study the case when $M$ …
Prime and Semiprime Ideals in Commutative Ternary $Γ$-Semirings: Quotients, Radicals, Spectrum
本文研究了交换三元Γ半环中的素理想与半素理想，通过商结构和谱拓扑揭示了新的代数结构。
In this paper we study and investigate concerning dependent elements of semiprime rings and prime rings R by using generalized derivation and derivation,when R admsit to satisfy some conditions,we give some results about that.
The $X$-semiprimeness of Rings
本文引入并研究环的 $X$-半素性，给出 $L$-半素判定准则，刻画素环的相关 Lie 理想，并分析多类环结构的性质。
In this paper we prove our main theorem, namely, theorem (8), which states that a link Q\rightarrowP, of prime ideals Q and P of a noetherian ring R that are σ-semistable with respect to a fixed automorphism σ of R, induces a link Q0\rightarrowP0 of the semiprime ideals Q0 and P0 of the ring R,where Q0 and P0 are the largest σ- invariant or σ- stable ideals contained in the prime ideals Q and P. We also prove a converse to this theorem.
Necessary and sufficient conditions are given for the completed group algebras of a compact p-adic analytic group with coefficient ring the p-adic integers or the field of p elements to be prime, semiprime and a domain. Necessary and sufficient conditions for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups are found. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coates et al.
The main purpose of this paper is to study and investigate concerning a (α,α)-symmetric derivations D on semiprime rings and prime rings R, we give some results when R admits a (α,α)-symmetric derivations D to satisfy some conditions on R.Where α: R to R is an automorphism mapping.
Prime and semiprime Lie ideals in C*-algebras
本文利用Dixmier理想理论，将C*-代数中的半素与素李理想分别刻画为半素与素双边理想的完全正规化子子空间，并在无特征含单位情形建立自然双射，给出完全非中心的等价条件。
Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules
本文证明任意交换环上任意模的半素子模与实子模，分别是包含它们的素子模与实素子模的交集，推广了此前仅适用于有限生成模的相关结论。
Prime and semiprime quantum linear space smash products
本文研究量子线性空间玻色子化与结合代数的 smash 积，给出其为（半）素的条件，并应用于量子仿射空间，推广了 Bergen 的结果。
Given a semiprime Goldie module $M$ projective in $σ[M]$ we study decompositions on its $M$-injective hull $\hat{M}$ in terms of the minimal prime in $M$ submodules. With this, we characterize the semiprime Goldie modules in $\mathbb{Z}$-Mod and make a decomposition of the endomorphism ring of $\hat{M}$. Also, we investigate the relations among semiprime Goldie modules, $QI$-modules and co-semisimple modules extending results on left $QI$-rings and $V$-rings.
The minimal primes of localizations of rings
本文研究环的局部化的极小素理想，在若干自然假设下给出其描述，并应用于半素环的最大商环，同时给出映射到中心的判据。
In this work, we extend the definition of the graded prime ideals from those in commutative graded rings to the ideals over graded Lie algebras. We prove some facts about graded prime Lie ideals in arbitrary Lie algebras that are similar to those about graded prime ideals over a commutative or non-commutative ring.In addition, the ideas of graded semiprime Lie ideals and graded total prime Lie ideals will be introduced.
The union of an ascending chain of prime ideals is not always prime. We show that this property is independent of the parallel property for semiprimes. We also show that the PI-class is a tight bound on the number of non-prime unions of subchains in a chain of primes in a PI-algebra.
Let $H$ be a cocommutative Hopf algebra acting on an algebra $A$. Assuming the base field to be algebraically closed and the $H$-action on $A$ to be integral, that is, it is given by a coaction of some Hopf subalgebra of the finite dual $H^\circ$ that is an integral domain, we stratify the prime spectrum $\mbox{Spec}\, A$ in terms of the prime spectra of certain commutative algebras. For arbitrary $H$-actions in characteristic $0$, we show that the largest $H$-stable ideal of $A$ that is contained in a given semiprime ideal of $A$ is semiprime as well.
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.…
Using the concepts of prime module, semiprime module and the concept of ascending chain condition (ACC) on annihilators for an $R$-module $M$ . We prove that if \ $M$ is semiprime \ and projective in $σ\left[ M\right] $, such that $M$ satisfies ACC on annihilators, then $M$ has finitely many minimal prime submodules. Moreover if each submodule $N\subseteq M$ contains a uniform submodule, we prove that there is a bijective correspondence between a complete set of representatives of isomorphism classes of indecomposable non $M$-singular injective modules in $σ\left[ M\right] $ and the set of min…
For every commutative differential algebra one can define the Lie algebra of special derivations. It is known for years that not every Lie algebra can be embedded to the Lie algebra of special derivations of some differential algebra. More precisely, the Lie algebra of special derivations of a commutative algebra always satisfy the standard Lie identity of degree $5$. The problem of existence of such embedding is a long-standing problem (see \cite{YuPSt5,PoinsotAAM,Poinsot}), which is closely related to the Lie algebra of vector fields on the affine line (see \cite{YuPSt5}). It was solved by R…
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.
In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let $H$ be a semisimple Hopf algebra over a field $\mathbb{k}$ and let $A$ be a left partial $H$-module algebra. We study the $H$-prime and the $H$-Jacobson radicals of $A$ and its relations with the prime and the Jacobson radicals of $\underline{A\#H}$, respectively. In particular, we prove that if $A$ is $H$-semiprimitive, then $\underline{A\#H}$ is semiprimitive provided that all irreducible representations of $A$ are finite-dimensional, or $A$ is an affine PI-algebra over $\mathbb{k}$ and $\…
Structure and Spectral Theory of Non-Commutative and $n$-ary $Γ$-Semirings
本文针对非交换与n元Γ-半环，拓展既有交换三元框架，定义多类理想、构建根理论与谱拓扑，最终得到非交换Wedderburn-Artin型分解，统一不同元数情形的相关理论。
Semiprime ideals in C*-algebras
本研究证明C*-代数中未必闭的理想是半素理想的充要条件为幂等或对正元平方根封闭，素与半素理想自动自伴；通过提出并研究Dixmier理想及其幂根理论，利用其对C*-代数理想的夹逼性质完成证明。
Prime rings having nontrivial centralizers of (skew) traces of Lie ideals
本文研究了带对合的素环中非交换 Lie 理想的迹与斜迹在非零导子作用下为零的性质，证明了此时环的迹或斜迹的平方包含于中心，并将结论推广至除环和含幂等元的素环的特定子群，推广了近期相关成果。
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…
It is known that every nonzero Jordan ideal of $2$-torsion free semiprime rings contains a nonzero ideal. In this paper we show that also any square closed Lie ideal of a $2$-torsion free prime ring contains a nonzero ideal. This can be interpreted by saying that studying identities over one sided ideals is the "optimal" case to study identities. With this fact in mind, we generalize some results of Dhara, Rehman and Raza in [Lie ideals and action of generalized derivations in rings, Miskolc Mathematical Notes, \textbf{16} (2015), 769 -- 779] to the context of nonzero left ideals.
We present two criteria for a group $G$ to satisfy the following statements: any $G$-graded gr-prime (gr-semiprime) right gr-Goldie ring admits a gr-semisimple graded right classical quotient ring. The criterion for gr-semiprime rings is that the group $G$ is periodic. Actually, the sufficiency of periodicity was proved by the author in 2011 and the necessity of it follows from the well-known counterexample (1979). The main result of the paper concerns the gr-prime case. In this case, Goodearl and Stafford proved the graded version of Goldie's Theorem for rings graded by an Abelian group (2000…
Certain functional identities involving a pair of homogeneous derivations with central values in gr-prime rings
本文研究了gr-素环中涉及一对同次导出的函数恒等式，证明了在特定条件下环的交换性，并指出结果无法推广到gr-半素环。
Prime Radical and Primary decomposition of Ideals in an L subring
本文在L-环中引入理想的素根、准素分解与约化准素分解概念，证明素根与根及半素根一致，并给出理想具有准素分解的充要条件。
It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) $R$ is right self-injective if $R_R$ is invariant under automorphisms of its injective hull. This answers two questions raised by Singh and Srivastava, and Clark and Huynh. An example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption. Also shown is that automorphism-invariant modules are precisely pseudo-injective modules, answering a recent question of Lee and Zhou. Furthermore, rings whose cyclic modules are automorphi…
Generalized Homogeneous Derivations on Graded Rings
本文探讨了广义齐次导出在分层环中的扩展，研究了gr-广义导出的性质，并将素环中的重要结果推广到gr-素环中，同时探讨了这些映射的代数与模结构及范畴框架。
On $z$-ideals and $z$-closure operations of semirings, I
本文研究半环的$z$-理想，引入并分析了$z$-素、$z$-半素等几类特殊理想及其性质，利用$z$-闭包算子证明了它们与相应理想类的等价性。
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 …
Counting Schur Rings over Cyclic Groups of Semi-prime Order
本文继续研究循环群上的Schur环计数问题，重点分析半素数阶循环群pq及4p阶循环群的结构与性质。
On ideals in quantales -- I
本文以环论为动机，系统研究交换quantale中理想的素性、半素性、根性、准素性、不可约性等关键性质，并探讨准素分解问题，为建立全面的quantale理想理论奠定基础。
Jacobson program for Lie algebras
本文借鉴环的结构理论，引入李模概念刻画本原李代数，证明表示定理与稠密性定理，并揭示核理想对素和半素李代数的判定作用。
On the Subtractive Ideal Structure of Commutative Semirings
本文研究了交换半环中减法性质对理想结构的影响，揭示了k-素和k-半素理想的新特征化，并证明了相关理想类别的重合性。
Revisiting Baer elements
本文将环的$d$-理想与框架的$d$-元素性质推广至乘法格中的Baer元素，提出了新结果，并引入Baer闭包探讨极大、素、半素与交不可约元。
We describe a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category. This provides a noncommutative version of Balmer's prime spectrum of a tensor triangulated category. These notions are based on containment conditions in terms of thick subcategories of an abelian category and thick ideals of an abelian 2-category. We prove categorical analogs of the main properties of noncommutative prime spectra. Similar notions, starting with Serre subcategories of an abelian category and Serre ideals of an abelian 2-category, are developed. They are linked…
Let R be a prime ring of H a generalized derivation and L a noncentral lie ideal of R. We show that if l^sH(l)l^t in Z(R) for all lin2 L, where s, t> 0 are fixed integers, then H(x) = bx for some b in C, the extended centroid of R, or R satisfies S4. Moreover, let R be a 2-torsion free semiprime ring, let A = O(R) be an orthogonal completion of R and B = B(C) the Boolean ring of C. Suppose ([x1; x2]sH([x1; x2])[x1; x2]t in Z(R) for all x1; x2 in R, where s, t> 0 are fixed integers. Then there exists idempotent e in B such that H(x) = bx on eA and the ring (1-e)A satisfies S4.
Reduced rank in $σ[M]$
本文研究了在σ[M]中降秩的概念扩展，探讨了商范畴的谱性质，并给出了模块商在特定torsion理论下的有限长度条件，以及推广Small定理的条件。
Regarding the question of how idempotent elements affect reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce {\it right} (resp., {\it left}) {\it $e$-reversible rings}. We show that this concept is not left-right symmetric. Basic properties of right $e$-reversibility in a ring are provided. Among others it is proved that if $R$ is a semiprime ring, then $R$ is right $e$-reversible if and only if it is right $e$-reduced if and only if it is $e$-symmetric if and only if it is right $e$-semicommutative. Also, for a right $e…
Left Jacobson Rings
本文研究了左雅各布环的性质，证明了多项式环的强左雅各布性及最大左理想的有限余维数，并探讨了Azumaya代数与中心的关系。
Ore Localisation for differential graded rings; Towards Goldie's theorem for differential graded algebras
本文研究微分分次代数的 Ore 局部化，定义 dg-素环、dg-半素环、dg-幂零根、dg-本质子模与 dg-一致维数，并推广 Goldie 定理至素 dg-环。
In this paper, we study rings having the property that every right ideal is automorphism-invariant. Such rings are called right $a$-rings. It is shown that (1) a right $a$-ring is a direct sum of a square-full semisimple artinian ring and a right square-free ring, (2) a ring $R$ is semisimple artinian if and only if the matrix ring $\mathbb{M}_n(R)$ for some $n>1$ is a right $a$-ring, (3) every right $a$-ring is stably-finite, (4) a right $a$-ring is von Neumann regular if and only if it is semiprime, and (5) a prime right $a$-ring is simple artinian. We also describe the structure of an indec…
Let $p$ be a an odd prime and let $G$ be a finite $p$-group with cyclic commutator subgroup $G'$. We prove that the exponent and the abelianization of the centralizer of $G'$ in $G$ are determined by the group algebra of $G$ over any field of characteristic $p$. If, additionally, $G$ is $2$-generated then almost all the numerical invariants determining $G$ up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of $G'$ is determined. These claims are known to be false for $p=2$.
Commutators and products of Lie ideals of prime rings
本文研究素环中李理想的结构，推广了 Herstein 经典定理，并完全刻画了李理想乘积包含非零理想及换位子为零的条件。
Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of extensions of $R[t]$-modules. Using this theory, we describe the similarity classes in $M_n(R)$ for $n\leq 4$, along with their centralizers. Among these, we characterize those classes which are similar to their transposes. Non-self-transpose classes are shown to exist for all $n>3$. When $R$ has finite residue field of order $q$, we enumerate the similarit…
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