M. A. González-Barroso, E. Pardo
In this note, we study the notion of purely infinite simple ring in the case of non-unital rings, and we obtain an analog to Zhang's Dichotomy for $σ$-unital purely infinite simple C*-algebras in the purely algebraic context.
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We gather some classical results and examples that show strict inclusion between the families of unital rings, rings with enough idempotents, rings with sets of local units, locally unital rings, s-unital rings and idempotent rings.
We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizi…
Let $\mathfrak{A}$ be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Jordan $n$-derivation $Δ:\mathfrak{A}\rightarrow\mathfrak{A}$ is additive. More precisely, it is proved that $Δ$ is of the form $Δ(t)=μt+δ(t),$ where $μ\in\mathcal{Z}(\mathfrak{A})$ and $δ:\mathfrak{A}\rightarrow\mathfrak{A}$ is a Jordan $n$-derivation. The main result is then applied to some classical examples of unital rings with nontrivial idempotents such as triangular rings, matrix rings, prime rings, nest algebras, standard operator a…
In this article, we study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;δ]$, under the hypothesis that $R$ is $s$-unital and $\ker(δ)$ contains a nonzero idempotent. This result generalizes a result by Öinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.
Let R be a commutative ring with unity and a let A be a not necessarily commutative R-algebra which is free as an R-module. If I is an ideal in A, one can ask when A/I is also free as an R-module. We show that if A has an admissible system and I has a unital Grobner basis then A/I is free as an R-module. We prove a version of Buchberger's theorem over R and, as a corollary, we obtain a Grobner basis proof of the Poincare-Birkhoff-Witt Theorem over a commutative ground ring.
Let $E$ be an arbitrary (countable) graph and let $R$ be a unital commutative ring. We analyze the ideal structure of the Leavitt path algebra $\lr$ introduced by Mark Tomforde. We first modify the definition of basic ideals and we then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of $\lr$. Then by applying these results we determine prime graded basic ideals and left (or right) primitive graded ideals of $\lr$. In particular, we show that when $E$ satisfies Condition (K) and $R$ is a field, the set …
When does an infinite ring have a finite compressed commuting graph?
本研究证明任意无限环都有无限非单位压缩交换图，以环的半直积为工具分类了具有有限单位压缩交换图的无限单位环，并由此分类了仅含有限个单位子环的无限单位环。
Invertibility of structured perturbations of singular matrices over unital rings
本文研究单位环上奇异矩阵经结构化扰动后的可逆性，证明在核与像的自然假设下可逆性等价于扰动矩阵可逆，并给出逆的显式公式，推广了Eriksson和Nordqvist的定理。
Artinian rings which are not generalized Rickart
该研究通过构造反例证明存在非单位右Artinian环不满足广义Rickart环性质，指出文献[16]的推论2.31在非单位Artinian环情形下不成立。
It is well-known that the Jacobson radical of a unital ring is its largest superfluous right ideal. It is recorded here that the result carries over to non-unital rings provided the notion of "superfluous" is taken relative to all regular ideals.
Given an extension $R \subseteq S$ of rings with same set of local units, inspired by the works of Miyashita, we construct four exact sequences of groups relating Picard's groups of $R$ and $S$.
Leavitt Path Algebras with Coefficients in a Commutative Unital Ring
本文将 Leavitt 路径代数的若干结果从域系数推广到交换单位环系数，并证明理想格嵌入、构造多项式增长的新基以及给出 Gelfand-Kirillov 维数的有向图公式。
Prime group graded rings with applications to partial crossed products and Leavitt path algebras
本文研究了素数群格环的性质，并将其应用于部分交叉积和莱维特路径代数中，扩展了相关经典结果。
In this article, we work with unital twisted partial action of $\mathbb{Z}$ on an unital ring $R$ and we introduce the twisted partial skew power series rings and twisted partial skew Laurent series rings. We study primality, semi-primality and prime ideals in these rings. We completely describe the prime radical in partial skew Laurent series rings. Moreover, we study the Goldie property in partial skew power series rings and partial skew Laurent series rings and we describe conditions of the semiprimality of twisted partial skew power series rings.
Rank of elements of general rings in connection with unit-regularity
该研究定义了一般含单位元环中元素的秩，给出半素环中秩的可逆元刻画，并证明含单位元半素环基座的元素均为单位正则元。
In a previous article (see \cite{CNP}), we introduced and analyzed ring-theoretic properties of object unital $\mathcal{G}$-graded rings $R$, where $\mathcal{G}$ is a groupoid. In the present article, we analyze the category $\grmod$ of unitary $\G$-graded modules over such rings. Following ideas developed earlier by one of the authors in \cite{lundstrom2004}, we analyze the forgetful functor $U \colon \grmod \to \rmod$ and aim to determine properties $\mathcal{P}$ for which the following implications are valid for modules $M$ in $\grmod$: $M$ is $\mathcal{P}$ $\Rightarrow$ $U(M)$ is $\mathcal…
Let $A$ be any unital associative, possibly non-commutative ring and let $p$ be a prime number. Let $E(A)$ be the ring of $p$-typical Witt vectors as constructed by Cuntz and Deninger and $W(A)$ be the abelian group constructed by Hesselholt. In arXiv:1708.04065 it was proved that if $ p=2$ and $A$ is non commutative unital torsion free ring then there is no surjective continuous group homomorphism from $W(A) \to HH_0(E(A)): = E(A)/\overline{[E(A),E(A)]}$ which commutes with the Verschiebung operator and the Teichmüller map. In this paper we generalise this result to all primes $p$ and simplif…
Tensor-Hom formalism for modules over nonunital rings
本文建立非单位环上模的张量-Hom形式体系，定义t-单位与c-单位环及模，刻画其范畴性质、与s-单位概念的关系及封闭性，证明两类模范畴的阿贝尔性与等价性，还探讨相关同调性质与数乘限制下的表现。
We begin by investigating the class of commutative unital rings in which no two distinct elements divide the same elements. We prove that this class forms a finitely axiomatizable, relatively ideal distributive quasivariety, and it equals the quasivariety generated by the class of integral domains with trivial unit group. We end the paper by proving a representation theorem that provides more evidence to the conjecture that Bézout monoids describe exactly the monoids of finitely generated ideals of commutative unital rings with distributive ideal lattice.
We show that if $R$ is a, not necessarily unital, ring graded by a semigroup $G$ equipped with an idempotent $e$ such that $G$ is cancellative at $e$, the non-zero elements of $eGe$ form a hypercentral group and $R_e$ has a non-zero idempotent $f$, then $R$ is simple if and only if it is graded simple and the center of the corner subring $f R_{eGe} f$ is a field. This is a generalization of a result of E. Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity o…
In the last 60 years coding theory has been studied a lot over finite fields $\mathbb{F}_q$ or commutative rings $\mathcal{R}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring with unity) of order $p^2$ had been presented, the construction of codes over non-commutative rings or non-commutative non-unital rings surfaced merely two years ago. In this letter, we extend the diverse research on exploring the codes over the non-commutative and non-unital ring $E= \langle 2a=2b=0, a^2=a, b^2=b, ab=a, ba=b \rangle$ by presenting the classi…
Given any finitely aligned higher-rank graph $Λ$ and any unital commutative ring $R$, the Kumjian-Pask algebra $\mathrm{KP}_R(Λ)$ is known as the higher-rank generalization of Leavitt path algebras. After characterizing simple Kumjian-Pask algebras by L.O. Clark and Y.E.P. Pangalela (and others), we focus in this article on the purely infinite simple ones. Briefly, we show that if $\mathrm{KP}_R(Λ)$ is simple and every vertex of $Λ$ is reached from a generalized cycle with an entrance, then $\mathrm{KP}_R(Λ)$ is purely infinite. We next prove a standard dichotomy for simple Kumjian-Pask algebr…
Let $R$ denote the purely infinite simple unital Leavitt path algebra $L(E)$. We completely determine the pairs of positive integers $(c,d)$ for which there is an isomorphism of matrix rings $M_c(R)\cong M_d(R)$, in terms of the order of $[1_R]$ in the Grothendieck group $K_0(R)$.
Let $R$ be a ring with a set of local units, and a homomorphism of groups $\underlineΘ : \G \to \Picar{R}$ to the Picard group of $R$. We study under which conditions $\underlineΘ$ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension $R \subseteq S$ with the same set of local units and assuming that $\underlineΘ$ is induced by a homomorphism of groups $\G \to \Inv{R}{S}$ to the group of all invertible $R$-sub-bimodules of $S$, then we construct an analogue of the Chase-Harrison-Rosenberg seven terms exac…
Jordan homomorphisms of triangular algebras over noncommutative algebras
本文将Benkovič关于交换环上三角矩阵代数Jordan同态的刻画推广到非交换环，并去除加法挠假设，证明了T_n(R)的Jordan同态限制在子代数T_n^0(R)上必为标准形式。
We study the relations between partial and global group cohomology with values in a commutative unital ring $\mathcal{A}$. In particular, for a unital partial action of a group $G$ on $\mathcal{A}$, such that $\mathcal{A}$ is a direct product of commutative indecomposable rings, we show that any partial $n$-cocycle of $G$ with values in $\mathcal{A}$ is globalizable.
Ring theory in o-minimal structures
本文在o-极小设定下发展了一般环理论，描述了所有可定义环，揭示了可定义连通环与有限维结合代数的强对应关系，并给出了非幺环嵌入可定义幺环的条件。
The covering number of an associative ring $R$ is the minimal number of proper subrings whose union is $R$. We establish a strategy to classify unital rings of a given finite covering number, and obtain a classification of unital rings whose covering number is four. Along the way we compute the covering number of every finite local ring whose residue field has prime order.
Let T be an n by n zero-square matrix over a commutative unital ring R. We show that T is similar to a multiple of E_1n if R is a GCD domain and n = 2, if R is a GCD domain with 2 not zero divisor and n = 3, but there are matrices which are not similar to any multiple of E_1n whenever n greater or equal 4, over any commutative unital ring.
Commutators of finite multiplicative order
本文研究复数矩阵上交换子方程[A,B]^k=Id_n，通过Lam和Leung的经典结果确定解的存在条件，并推广到任意单位环上的矩阵环，进一步探讨一般单位环中交换子方程[a,b]^n=1的结构影响，证明在特定条件下该方程等价于环同构于某矩阵环。
Let $R$ be a unital ring with involution.In this paper, several new necessary and sufficient conditions for the existence of the Moore-Penrose inverse of an element in a ring $R$ are given.In addition, the formulae of the Moore-Penrose inverse of an element in a ring are presented.
Separability for relative extensions of object unital strongly groupoid graded rings
本文研究了在群oids格环的相对扩展下，环R/R_Δ的分离性条件，通过引入相对跟踪映射和中心元素，推广了关于矩阵环、群格环以及群oids格环的分离性定理。
In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-)unital $K$-algebra having enough nonzero idempotents (for example, for a vo…
Cancellation properties of graded and nonunital rings. Graded clean and graded exchange Leavitt path algebras
本文将单位环的消去相关性质推广到非单位环和分次环，研究分次clean与分次exchange在分次矩阵环下的行为，并刻画Leavitt路径代数为分次clean或分次exchange的图论条件。
Let $\Mn$ be the ring of all $n \times n$ matrices over a unital ring $\mathcal{R}$, let $\mathcal{M}$ be a 2-torsion free unital $\Mn$-bimodule and let $D:\Mn\rightarrow \mathcal{M}$ be an additive map. We prove that if $D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)=0$ whenever $\A,\B\in \Mn$ are such that $\A\B=\B\A=0$, then $D(\A)=δ(\A)+\A D(\textbf{1})$, where $δ:\Mn\rightarrow \mathcal{M}$ is a derivation and $D(\textbf{1})$ lies in the centre of $\mathcal{M}$. It is also shown that $D$ is a generalized derivation if and only if $D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)-\A D(\textbf{1})\B-\B D(\textbf{1…
The purpose of this short note is to prove that if $A$ and $B$ are unital C*-algebras and $ϕ: A \to B$ is a unital *-preserving ring homomorphism, then $ϕ$ is contractive; i.e., $\| ϕ(a) \| \leq \| a \|$ for all $a \in A$. (Note that we do not assume $ϕ$ is linear.) We use this result to deduce a number of corollaries as well as characterize the form of such unital *-preserving ring homomorphisms. (This note may be of interest to C*-algebraists as well as algebraists who study noncommutative rings and algebras. It is meant to be accessible to a general mathematician and does not require any pr…
Very good gradings on matrix rings are epsilon-strong
本文研究矩阵环上的群分次，引入非常好分次并证明其必为 epsilon-强，同时给出 epsilon-交叉积的充分条件及 IBN 情形下的刻画。
Let $R$ be a unital ring with involution. We give several characterizations and properties of core partial order in $R$. In particular, we investigate the reverse order law $(ab)^{\tiny\textcircled{\tiny\#}} = b^{\tiny\textcircled{\tiny\#}} a^{\tiny\textcircled{\tiny\#}}$ for two core invertible elements $a,b\in R$. Some relationships between core partial order and other partial orders are obtained.
We prove that Jordan triple elementary surjective maps on unital rings containing a nontrivial idempotent are automatically additive.
In this paper we work with unital twisted partial actions. We investigate ring theoretic properties of partial crossed products as artinianity, noetherianity, perfect property, semilocalproperty, semiprimary property and we also study the Krull dimension. Moreover, we consider triangular matrix representation of partial skew group rings, weak and global dimensions of partial crossed products Also we study when the partial crossed products are Frobenius and symmetric algebras.
Artinian Meadows
本研究提出阿廷草域的代数结构概念，证明其可本质唯一分解为局部草域乘积，并给出从带单位元交换环构造草域的规范方法，丰富了代数结构的分解理论。
A Note on Quaternions over Commutative Rings
本文在交换幺环上定义四元数，推广其若干性质，并研究单位群，特别关注halidon环。
We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if $R$ is a purely infinite simple ring, then $K_0(R)^+= K_0(R)$, the monoid of isomorphism classes of finitely generated projective $R$-modules is isomorphic to the monoid obtained from $K_0(R)$ by adjoining a new zero element, and $K_1(R)$ is the abelianization of the group of units of $R$. We develop techniques of construction, obtaining new examples in this class in the case of von Neumann regular rings, and we com…
Let Q be a (non-unital) simple ring. A nonempty subset S of Q is said to have zero product if S^2=0. We classify all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.
A cover of an associative (not necessarily commutative nor unital) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $σ(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $σ$-elementary if $σ(R) < σ(R/I)$ for every nonzero two-sided ideal $I$ of $R$. If $R$ is a ring with unity, then we define the unital covering number $σ_u(R)$ to be the size of a minimal cover of $R$ by subrings that contain $1_R$ (if such a cover exists), and $R$ is $σ_u$-elementary if $σ_u(R) < σ_u(R/I)$ for ever…
Given a non-necessarily commutative unital ring $R$ and a unital partial representation $Θ$ of a group $G$ into the Picard semigroup $\mathbf{PicS} (R)$ of the isomorphism classes of partially invertible $R$-bimodules, we construct an abelian group $\mathcal{C}(Θ/R) $ formed by the isomorphism classes of partial generalized crossed products related to $Θ$ and identify an appropriate second partial cohomology group of $G$ with a naturally defined subgroup $\mathcal{C}_0(Θ/R) $ of $\mathcal{C}(Θ/R).$ Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact seque…
Let $\mathcal A$ and $\mathcal B$ be unital rings and $\mathcal M$ be a $(\mathcal A, \mathcal B)$-bimodule, which is faithful as a left $\mathcal A$-module and also as a right $\mathcal B$-module. Let ${\mathcal U}={\rm Tri}(\mathcal A, \mathcal M, \mathcal B)$ be the associated triangular ring. It is shown that every additive generalized Jordan (triple) higher derivation on $\mathcal U$ is a generalized higher derivation.
On nonmatrix varieties of associative rings
本文研究了结合环的非矩阵范畴，推广了已知结果到一般环的情况，并扩展到不包含矩阵代数的范畴。
Let $S$ be a unital ring, $S[t;σ,δ]$ a skew polynomial ring where $σ$ is an injective endomorphism and $δ$ a left $σ$-derivation, and suppose $f\in S[t;σ,δ]$ has degree $m$ and an invertible leading coefficient. Using right division by $f$ to define the multiplication, we obtain unital nonassociative algebras $S_f$ on the set of skew polynomials in $S[t;σ,δ]$ of degree less than $m$. We study the structure of these algebras. When $S$ is a Galois ring and $f$ base irreducible, these algebras yield families of finite unital nonassociative rings $A$, whose set of (left or right) zero divisors has…
Symplectic Quasi Self-dual, Self-dual, and LCD Codes over Non-unital Rings of Order Six
本文研究两个六阶非单位环上的辛码，通过二元与三元分量分解刻画自正交、自对偶、拟自对偶及LCD码，并利用自同构群和双陪集枚举完成短偶长度码的置换等价分类。
Categorial properties of compressed zero-divisor graphs of finite commutative rings
该研究定义了有限交换含幺环基于相伴关系的压缩零因子图$\u0398(K)$，证明其为诱导函子的最优压缩且函子保持范畴积，并借助该图结构刻画了局部环、主理想环等重要环类。
The perturbation of the group inverse under the stable perturbation in a unital ring
本文研究单位环中在稳定扰动下群逆的扰动，给出了扰动后群逆的显式表达式，推广了Xue(2007)、Xue和Chen(2007)以及Xue(2012)的相关结果，并应用于环上特定矩阵的群逆表示。
Let $ξ:C^*(E)\to C^*(F)$ be a unital $*$-homomorphism between simple purely infinite Cuntz-Krieger algebras of finite graphs. We prove that there exists a unital $*$-homomorphism $ϕ:L(E)\to L(F)$ between the corresponding Leavitt path-algebras such that $ξ$ is homotopic to the map $\hatϕ:C^*(E)\to C^*(F)$ induced by completion. We show moreover that $\hatϕ$ is a homotopy equivalence in the $C^*$-algebraic sense if and only if $ϕ$ is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz-Krieger algebras is …
Root graded groups revisited
本文研究根分级群的代数结构，描述了不可约根系中rank≥3（排除H3和H4）的变种，并提供其构造方法。
Let $n\ge 2$, let $\mathcal{R}_n$ be the graph consisting of one vertex and $n$ loops and let $\mathcal{R}_{n^-}$ be its Cuntz splice. Let $L_n=L(\mathcal{R}_n)$ and $L_{n^-}=L(\mathcal{R}_{n^-})$ be the Leavitt path algebras over a unital ring $\ell$. Let $C_m$ be the cyclic group on $2\le m\le \infty$ elements. Equip $L_n$ and $L_{n^-}$ with their natural $C_m$-gradings. We show that under mild conditions on $\ell$, which are satisfied for example when $\ell$ is a field or a PID, there are no unital $C_m$-graded ring homomorphisms $L_n\to L_{n^-}$ nor in the opposite direction.
The mod-p cohomology ring of a non-trivial finite p-group is an infinite dimensional, finitely presented graded unital algebra over the field with p elements, with generators in positive degrees. We describe an effective algorithm to test if two such algebras are graded isomorphic. As application, we determine all graded isomorphisms between the mod-p cohomology rings of all p-groups of order at most 100.
We consider a generalization of the quaternion ring $\Big(\frac{a,b}{R}\Big)$ over a commutative unital ring $R$ that includes the case when $a$ and $b$ are not units of $R$. In this paper, we focus on the case $R=\mathbb{Z}/n\mathbb{Z}$ for and odd $n$. In particular, for every odd integer $n$ we compute the number of non-isomorphic generalized quaternion rings $\Big(\frac{a,b}{\mathbb{Z}/n\mathbb{Z}}\Big)$
Compressed zero-divisor graphs of noncommutative rings
本研究将压缩零因子图概念推广至非交换环，构造保积函子，刻画有限域上矩阵环对应图的性质与自同构，证明满足条件的环同构可由其压缩零因子图同构推导。
Extensions of Centrally Essential Rings
本文通过理想扩张、Dorroh扩张和平凡扩张描述了几乎完全素中心本质环的结构。
Let $A$ and $B$ be unital rings. An additive map $T:A\to B$ is called a weighted Jordan homomorphism if $c=T(1)$ is an invertible central element and $cT(x^2) = T(x)^2$ for all $x\in A$. We provide assumptions, which are in particular fulfilled when $A=B=M_n(R)$ with $n\ge 2$ and $R$ any unital ring with $\frac{1}{2}$, under which every surjective additive map $T:A\to B$ with the property that $T(x)T(y)+T(y)T(x)=0$ whenever $xy=yx=0$ is a weighted Jordan homomorphism. Further, we show that if $A$ is a prime ring with char$(A)\ne 2,3,5$, then a bijective additive map $T:A\to A$ is a weighted Jo…
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