L. Oukhtite, S. Salhi, L. Taoufiq
Let $R$ be a 2-torsion free semiprime ring and $U$ a nonzero square closed Lie ideal of $R$. In this paper it is shown that if $f$ is either an endomorphism or an antihomomorphism of $R$ such that $f(U)=U,$ then $f$ is strong commutativity preserving on $U$ if and only if $f$ is centralizing on $U.$
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In this paper, firstly as a short note, we prove that a left derivation of a semiprime $Γ$-ring $M$ must map $M$ into its center, which improves a result by Paul and Halder and some results by Asci and Ceran. Also we prove that a semiprime $Γ$-ring with a strong commutativity preserving derivation on itself must be commutative and that a strong commutativity preserving endomorphism on a semiprime $Γ$-ring $M$ must have the form $σ(x)=x+ζ(x)$ where $ζ$ is a map from $M$ into its center, which extends some results by Bell and Daif to semiprime $Γ$-rings.
In this paper, we introduce multiplicative semiderivation and we investigate the commutativity of semiprime rings satisfying certain conditions and identities involving multiplicative semiderivations on a nonzero ideal I of a ring R.
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.
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…
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.
In this paper we investigate the Lie structure of the Lie superalgebra K of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U is a Lie ideal of K, then either there exists an ideal J of A such that a determined nonzero Lie ideal connected with J is contained in U, or A is a subdirect sum of A', A'', where the image of U in A' is central, and A'' is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center.
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。
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。
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.
In this paper we investigate the Lie structure of the derived Lie superalgebra [K, K], with K the set of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U is a Lie ideal of [K, K], then either there exists an ideal J of A such that the Lie ideal [J \cap K,K] is nonzero and contained in U, or A is a subdirect sum of A', A'', where the image of U in A' is central, and A'' is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center.
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.
$\textbf{Theorem 1.3.}$ For a given ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ Every non-singular right $A$-module $X$ which is is injective with respect to some essential right ideal of the ring $A$ is an injective module. $\textbf{2)}$ $A/G(A_A)$ is a right strongly semiprime ring. $\textbf{Theorem 1.4.}$ For a given ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right strongly semiprime ring. $\textbf{2)}$ Every right $A$-module which is injective with respect to some essential right ideal of the ring $A$, i…
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 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。
A ring $A$ is called Hopfian if $A$ cannot be isomorphic to a proper homomorphic image $A/J$. $A$ is called Bassian if there cannot be an injection of $A$ into a proper homomorphic image $A/J$. We consider classes of Hopfian and Bassian rings, and tie representability of algebras and chain conditions on ideals to these properties. In particular, any semiprime algebra satisfying the ACC on semiprime ideals is Hopfian, and any semiprime affine PI-algebra over a field is Bassian.
On derivations and semiprime ideal of rings
本文研究了在特定条件下的环导出与半素理想的关系，证明了在满足特定微分恒等式时导出的交换性质，并给出了半素理想变体的结果。
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.
In this paper we introduce the definition of a noetherian disjoint ring and that of a noetherian non-disjoint ring . For a noetherian ring R , with nilradical N if P and Q represent the semiprime ideals of R called as the right and the left krull-homogenous parts of N as defined in [8] , then we prove the main theorem of this paper for the ring R whose statement is given below. Main Theorem :- Let R be a Noetherian ring with nilradical N . Let P and Q represent the right and the left krull-homogenous parts of N . Then the following hold true for the ring R ; (a) If R is a disjoint ring , then …
In this paper we examine the commutativity of ideal extensions. We introduce methods of constructing such extensions, in particular we construct a noncommutative ring T which contains a central and idempotent ideal I such that T/I is a field. This answers a question from [2]. Moreover we classify fields of characteristic 0 which can be obtained as T/I for some T.
In order to simultaneously generalize matrix rings and group graded crossed products, we introduce category crossed products. For such algebras we describe the center and the commutant of the coefficient ring. We also investigate the connection between on the one hand maximal commutativity of the coefficient ring and on the other hand nonemptyness of intersections of the coefficient ring by nonzero twosided ideals.
We investigate properties of commutative subrings and ideals in non-commutative algebraic crossed products for actions by arbitrary groups. A description of the commutant of the base coefficient subring in the crossed product ring is given. Conditions for commutativity and maximal commutativity of the commutant of the base subring are provided in terms of the action as well as in terms of the intersection of ideals in the crossed product ring with the base subring, specially taking into account both the case of base rings without non-trivial zero-divisors and the case of base rings with non-tr…
In this paper for a noetherian ring R with nilradical N we define semiprime ideals P and Q called as the left and right krull homogenous parts of N . We also recall the known definitions of localisability and the weak ideal invariance (w.i.i for short ) of an ideal of a noetherian ring R . We then state and prove results that culminate in our main theorem whose statement is given below ; Theorem :- Let R be a noetherian ring with nilradical N . Let P and Q be semiprime ideals of R that are the right and left krull homogenous parts of N respectively . Then the following conditions are equivalen…
Let $R$ be a 2-torsion free $σ$-prime ring, $U$ a nonzero square closed $σ$-Lie ideal of $R$ and let $d$ be a derivation of $R$. In this paper it is shown that: 1) If $d$ is centralizing on $U$, then $d = 0$ or $U \subseteq Z(R)$. 2) If either $d([x, y]) = 0$ for all $x, y \in U$, or $[d(x), d(y)] = 0$ for all $x, y \in U$ and $d$ commutes with $σ$ on $U$, then $d = 0$ or $U \subseteq Z(R)$.
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.
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.
It is an open question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central $H$-invariant elements of the Martindale ring of quotients of a module algebra form a von Neumann regular and self-injective ring whenever $A$ is semiprime. For a semiprime Goldie PI $H$-module algebra $A$ with central invariants we show that $\AH$ is semiprime if and only if the $H$-action c…
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.
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。
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.
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 …
Affirmative answer to the Question of Leroy and Matczuk on injectivity of endomorphisms of semiprime left Noetherian rings with large images
本文对 Leroy 和 Matczuk 提出的关于具有大像的半素左 Noetherian 环自同态是否必为单射的问题给出了肯定回答，并证明了更一般的二分法定理及具有 Krull 维数的半素环的相关结论。
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.
This paper continues a research program on constructive investigations of non-commutative Ore localizations, initiated in our previous papers, and particularly touches the constructiveness of arithmetics within such localizations. Earlier we have introduced monoidal, geometric and rational types of localizations of domains as objects of our studies. Here we extend this classification to rings with zero divisors and consider Ore sets of the mentioned types which are commutative enough: such a set either belongs to a commutative algebra or it is central or its elements commute pairwise. By using…
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…
A partly (anti-)commutative quiver algebra is a quiver algebra bound by an (anti-)commutativity ideal, that is, a quadratic ideal generated by monomials and (anti-)commutativity relations. We give a combinatorial description of the ideals and the associated generator graphs, from which one can quickly determine if the ideal is admissible or not. We describe the center of a partly (anti-)commutative quiver algebra and state necessary and sufficient conditions for the center to be finitely generated as a K-algebra. As an application, necessary and sufficient conditions for finite generation of t…
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。
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。
Let A be a semprime, right noetherian ring equipped with an automorphism alpha, and let B := A[[y; alpha]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.
We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra $χ(\mathfrak{g})$ generated by two isomorphic copies $\mathfrak{g}$ and $\mathfrak{g}^ψ$ of a fixed Lie algebra, subject to the relations $[x,x^ψ]=0$ for all $x \in \mathfrak{g}$. In this article we study the ideal $L =L(\mathfrak{g})$ generated by $x-x^ψ$ for all $x \in \mathfrak{g}$. We obtain an (infinite) presentation for $L$ as a Lie algebra, and we show that in general it cannot be reduced to a finite one. With this in hand, we study the question of nilpotency. We show that if $\math…
Divide and Transfer: Non-Unique Factorizations Beyond Commutativity
本文研究非交换Dedekind素环中的非唯一分解问题，通过模论方法和图示计算探讨除数理论与转移同态的应用。
Quotient Modules of Finite Length and Their Relation to Fredholm Elements in Semiprime Rings
本文探讨了半素环中模长度与Fredholm元素的关系，提出基于有限长度商模的广义Fredholm元素概念，并证明了在含幺C*-代数中该条件与Fredholm性质等价。
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…
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。
Let R0 be a commutative associative ring (not necessarily unital), G a group and alpha a partial action by ideals that contain local units. We show that R0 is maximal commutative in the partial skew group ring R0*G if and only if R0 has the ideal intersection property in R0*G. From this we derive a criterion for simplicity of R0*G in terms of maximal commutativity and $G-$simplicity of R0 and apply this to two examples, namely to partial actions by clopen subsets of a compact set and to give a new proof of the simplicity criterion for Leavitt path algebras. A new proof of the Cuntz-Krieger uni…
We study the analogue of Sidki's weak commutativity construction, defined originally for groups, in the category of Lie algebras. This is the quotient $χ(\mathfrak{g})$ of the Lie algebra freely generated by two isomorphic copies $\mathfrak{g}$ and $\mathfrak{g}^ψ$ of a fixed Lie algebra by the ideal generated by the brackets $[x,x^ψ]$, for all $x$. We exhibit an abelian ideal of $χ(\mathfrak{g})$ whose associated quotient is a subdirect sum in $\mathfrak{g} \oplus \mathfrak{g} \oplus \mathfrak{g}$ and we give conditions for this ideal to be finite dimensional. We show that $χ(\mathfrak{g})$ h…
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。
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。
In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring $R = \bigoplus_{g\in G} R_g$ the grading group $G$ acts, in a natural way, as automorphisms of the commutant of the neutral component subring $R_e$ in $R$ and of the center of $R_e$. We show that if $R$ is a strongly $G$-graded ring where $R_e$ is maximal commutative in $R$, then $R$ is a simple ring if and only if $R_e$ is $G$-simple (i.e. there are no nontrivial $G$-invariant ideals). We also show that if $R_e$ is commutative (not necessarily maximal com…
We introduce a lattice structure as a generalization of meet-continuous lattices and quantales. We develop a point-free approach to these new lattices and apply these results to $R$-modules. In particular, we give the module counterpart of the well known result that in a commutative ring the set of semiprime ideals, that is, the set of radical ideals is a frame.
In the present paper, we investigate the commutativity of quotient ring $R/P$ where $R$ is any ring and $P$ is a prime ideal of $R$ which admits generalized derivations are satisfying some algebraic identities acting on prime ideals $P$.
Order-to-ring ideals via submultiplicative lattice seminorms in lattice-ordered algebras
本文刻画了具有序到环理想性质的格序代数，证明阿基米德格序代数具有该性质当且仅当它是f-代数且具有零半范数元素为幂零元的子乘法格半范数，并推广到半素f-代数的情形。
Structure of a factor ring $R/P$ in terms of differential identities involving a new kind of involution
本文研究商环 $R/P$ 的交换性，利用 $R$ 的导子所满足的、涉及与素理想 $P$ 相关的新型对合的代数恒等式，建立了若干交换性判据。
A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization…
Several authors have studied the question of when the monoid ring DM of a monoid M over a ring D is a right and/or left fir (free ideal ring), a semifir, or a 2-fir (definitions recalled in section 1). It is known that for M nontrivial, a necessary condition for any of these properties to hold is that D be a division ring. Under that assumption, necessary and sufficient conditions on M are known for DM to be a right or left fir, and various conditions on M have been proved necessary or sufficient for DM to be a 2-fir or semifir. A sufficient condition for DM to be a semifir is that M be a dire…
Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In the present paper we show that the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$ is generated as a group by the elements of the two following forms: 1) $z_{ij}(ab,c)$ and $z_{ij}(ba,c)$, 2) $[t_{ij}(a),t_{ji}(b)]$, where $1\le i\neq j\le n$, $a\in A$, $b\in B$, $c\in R$. Moreover, for the second type of generators, it suffices to fix one pair of indices $(i,j)$. This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies tha…
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