Artin环的覆盖定理

统一关于Artinian环与模的一些经典结果

Unifying some classical results on Artinian rings and modules

本文引入环R上左R-模类的测度概念，借此为Akizuki、Anderson、Hopkins和Levitzki关于Artinian环与模的经典定理及部分新结果提供了简短证明。

半素环嵌入半单阿廷环

Embeddings of semiprime rings into semisimple Artinian rings

本文给出半素环嵌入半单阿廷环的判据，引入并研究三类嵌入，证明最小嵌入为初等嵌入，并给出基于中心的新判据及最小嵌入的显式描述。

Artinian bimodule with quasi-Frobenius bimodule of translations

Let M be an (A,B)-bi-module over left and right respectively Artinian rings. Let C be a multiplication ring of bi-module M, i.e. the ring C is generated by images of a and B in End(M). Let Z be a center of C. Bi-module M over rings C and Z we call a bi-module of translation of (A,B)-bi-module M. We investigate a relationship between the fact that (A,B)-bi-module M is quasi-Frobenius and the fact that (C,Z)-bi-module M is quasi-Frobenius.

Artinian Meadows（阿廷草域）

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。

单Artinian环中序上的星运算

Star operation on orders in simple Artinian rings

星运算是乘法理想理论中的一个重要工具。本文利用一种特殊的星形运算，称为$ν$ -运算，定义了右Prüfer $ν$ -乘法阶的概念.后者可以被看作是Prüfer $ν$ -乘法域的自然非交换版本。作为我们的主要结果之一，我们建立了右Prüfer $ν$ -乘法阶的一个覆盖环也是右Prüfer $ν$ -乘法阶.

Filtered skew derivations on simple artinian rings

Given a complete, positively filtered ring $(R,f)$ and a compatible skew derivation $(σ,δ)$, we may construct its skew power series ring $R[[x;σ,δ]]$. Due to topological obstructions, even if $δ$ is an \emph{inner} $σ$-derivation, in general we cannot ``untwist" it, i.e. reparametrise to find a filtered isomorphism $R[[x; σ, δ]] \cong R[[x'; σ]]$, as might be expected from the theory of skew polynomial rings; similarly when $σ$ is an inner automorphism. We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of $R[[x;σ,δ]]…

每个右理想自同构不变的环

Rings with each right ideal automorphism-invariant

本文研究了具有每个右理想是自同构不变的性质的环。这种环称为右$a$ -环。证明了：（1）右$a$ -环是平方全半单Artin环与右平方自由环的直和;（2）环$R$是半单Artin环当且仅当某个$n>1$的矩阵环$\mathbb{M}_n(R)$是右$a$ -环;（3）右$a$ -环是稳定有限的，（4）右$a$ -环是von Neumann正则的当且仅当它是半素的;（5）素右$a$ -环是单Artin环。我们还把不可分解的右Artin右非奇异右$a$ -环的结构刻画为某些分块矩阵的三角形矩阵环。

Morphic与主理想群环

Morphic and principal-ideal group rings

我们观察到左和右Artin左和右morphic环类与Artin主理想环类一致。对$R$是Artin主理想环，$G$是群，刻画了$RG$是主理想环的情形;对有限群$G$，刻画了$RG$是左、右morphic环的情形.这扩展工作的Passman，Sehgal和费舍尔的情况下，当$R$是一个领域，和工作的陈，李，周的形态群环。

Artinian局部环理论的完全闭模型

Existentially closed models of the theory of Artinian local rings

所有长度不超过l的Artinian局部环的类是A_2-初等的，由一个有限集的公理Art_l公理化.我们证明了它的存在闭模型是Gorenstein，长度正好为l，并且它们的剩余域是代数闭的，并且，相反地，每个存在闭模型都是这种形式。所有具有代数闭剩余域的长度为l的Artinian局部Gorenstein环的理论Gor_l是模型完备的，而理论Art_l是可伴的，它具有模型伴Gor_l.

Semidistributive Laurent Series Rings

If $A$ is a ring with automorphism $φ$ and the skew Laurent series ring $A((x,φ))$ is a right semidistributive semilocal ring then $A$ is a right semidistributive right Artinian ring. The Laurent series ring $A((x))$ is a right semidistributive semilocal ring if and only if $A$ is a right semidistributive right Artinian ring. The study is supported by Russian Scientific Foundation (project 16-11-10013).

具有局部单位的无平方环

Square-Free Rings With Local Units

最近，作者用单位元刻划了所有的无artinian平方环。在这里，这些结果被扩展到设置的环与本地单位。我们使用这种特征的平方自由环，以获得许多结果类似的阿廷的情况下，并确定，在上同调方面，正是当一个平方自由环有相关的平方-自由代数结构。

自动生成的非artinian单片模块

Finitely generated, non-artinian monolithic modules

本文研究了任意单模的内射壳都是局部Artin的诺特环$A$。然后给出了不具有此性质的代数$A$的一般构造。在特征0中，我们也完成了具有此性质的down-up代数的分类。

Artinian and noetherian partial skew groupoid rings

Let $α= \{ α_g : R_{g^{-1}} \rightarrow R_g \}_{g \in \textrm{mor}(G)}$ be a partial action of a groupoid $G$ on a non-associative ring $R$ and let $S = R \star_α G$ be the associated partial skew groupoid ring. We show that if $α$ is global and unital, then $S$ is left (right) artinian if and only if $R$ is left (right) artinian and $R_g = \{ 0 \},$ for all but finitely many $g \in \textrm{mor}(G)$. We use this result to prove that if $α$ is unital and $R$ is alternative, then $S$ is left (right) artinian if and only if $R$ is left (right) artinian and $R_g = \{ 0 \},$ for all but finitely ma…

Artin模上的双线性映射

Bilinear maps on Artinian modules

证明了如果交换环k上模的双线性映射f：A x B --> C是非退化的（即，如果A中没有非零元素零化B的全部，反之亦然），且A和B是阿廷的，则A和B是有限长的。注意到一些后果。反例给出了一些尝试推广上述声明的平衡双线性映射的非交换环上的双模，而提出的问题是否其他这样的推广是真的。

On the problem of the finiteness of the compressed zero divisor graphs of Artinian rings

Let R be an Artinian ring and G be the compressed zero-divisor graph associated to R. The question of when the clique number of compressed zero-divisor graphs is finite was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff, in their survey paper entitled On Zero-divisor Graphs. They proved that if length of R is at most four then the clique number is finite. In the length six case they gave an example of a ring where the clique is infinite. In this paper we show that when length of ring is five then the clique number of compressed zero-divisor graph is finite.

Weak diamond, weak projectivity, and transfinite extensions of simple artinian rings

We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. We prove that if R is small, then the Weak Diamond implies that projectivity of an arbitrary module can be tested at the layer epimorphisms of R.

On cyclic essential extensions of simple modules over differential operator rings

In this paper we discuss under which conditions cyclic essential extensions of simple modules over a differential operator ring R[z;d] are Artinian. In particular, we study the case when R is either d-simple or d-primitive. Furthermore, we obtain important results when R is an affine algebra of Kull dimension 2. As an application we characterize the differential operator rings C[x,y][z;d] for which cyclic essential extensions of simple modules are Artinian.

微分算子环上单模的内射壳

Injective hulls of simple modules over differential operator rings

研究了微分算子环$R[θ; d]$上单模的内射壳，给出了这些模是局部Artin模的必要条件.因此，我们刻画了$S=K[x][θ;σ, d]$的Ore扩展，$d$是$K$ -线性导子，$σ$是$K[x]$的$K$ -线性自同构，使得简单$S$ -模的内射壳是局部Artin的.

Galois Corings与Jacobson-Bourbaki型对应

Galois Corings and a Jacobson-Bourbaki type Correspondence

关于除环的Jacobson-Bourbaki定理是由Sweedler在1975年用核环来表述的。在这种新方法中，不需要假设条件。在本文中，我们推广Sweedler的结果，简单的artinian环使用一类特殊的核，余矩阵核。利用对偶性得到了简单Artin环的Jacobson-Bourbaki型对应。

Covering classes and uniserial modules

We apply minimal weakly generating sets to study the existence of Add$(U_R)$-covers for a uniserial module $U_R$. If $U_R$ is a uniserial right module over a ring $R$, then $S:=$End$ (U_R)$ has at most two maximal (right, left, two-sided) ideals: one is the set $I$ of all endomorphisms that are not injective, and the other is the set $K $ of all endomorphisms of $U_R$ that are not surjective. We prove that if $U_R$ is either finitely generated, or artinian, or $I \subset K$, then the class Add$(U_R)$ is covering if and only if it is closed under direct limit. Moreover, we study endomorphism ri…

左Artin环的左局部化

Left localizations of left Artinian rings

对任意左Artinian环$R$，给出了$R$的所有左分母集$S$和$R$的左局部化$S^{-1}R$的显式描述.证明了在$R$ -同构之前，只有2个左局部化，且每个左局部化都是幂等局部化，即$S^{-1}R\simeq S_e^{-1}R$和${\rm ass} (S) = {\rm ass} (S_e)$，其中$S_e=\{1,e\}$是$R$的左分母集，$e$是幂等局部化.此外，幂等$e$直到共轭为止都是唯一的。证明了$R$的极大左分母集的个数是有限的，且不超过单左$R$ -模的同构类的个数.给出了$R$的极大左分母集和$R$的左局部化根的定义.

准射线猜想在艺术环上有向图表示的导出范畴中的应用

Telescope conjecture for quiver representations over artinian rings

本文证明了在艺术环上有限有向图表示的导出范畴中的准射线猜想，并扩展到交换完美环的非noetherian背景下。

分级链条件和群oid分级雅可比纯环的群oid分级模

Graded chain conditions and graded Jacobson radical of groupoid graded modules

本文研究了群oid分级模的分级链条件、分级雅可比纯环及gr-semilocal环，探讨了其性质与经典理论的异同。

Modules close to SSP- and SIP-modules

In this paper, we investigate some properties of SIP, SSP and CS-Rickart modules. We give equivalent conditions for SIP and SSP modules; establish connections between the class of semisimple artinian rings and the class of SIP rings. It shows that $R$ is a semisimple artinian ring if and only if $R_R$ is SIP and every right $R$-module has a SIP-cover. We also prove that $R$ is a semiregular ring and $J(R) = Z(R_R)$ if only if every finitely generated projective module is a SIP-CS module which is also a $C2$ module.

半artinian Von Neumann正则代数上artinian偏序集的表示

Representation of artinian partially ordered sets over semiartinian Von Neumann regular algebras

如果$R$是一个半阿丁冯诺依曼正则环，则$R$的本原理想的集合$\Prim_{R}$，按包含序，是一个阿丁偏序集，其中所有极大链都有一个最大元素。此外，如果$\Prim_{R}$没有无限反链，则$R$的所有理想的格$\BL_{2}(R)$与$\Prim_{R}$的所有上子集的格反同构。由于赋值$U\mapsto r_R(U)$定义了从单右$R$ -模的代表的任何集合$\Simp_R$到$\Prim_{R}$的双射，在$\Simp_R$中诱导了一个自然偏序，在该偏序下，最大元素恰好是那些在各自的自同态除环上是有限维的单右$R$ -模;这些总是$R$ -内射的。给出了一种构造方法，它构造了一个半阿丁单位正则的$D$ -代数$D_I$，它具有以下特征：（a）$\Simp_{D_I}$与$I$序同构;（b）赋值$H\mapsto\Simp_{D_I/H}$实现了从格$\BL_{2}(D_I)$到$\Simp_{D_I}$的所有上子集的格的反同构;（c）$\Simp_{D_I}$的一个非极大元是内射的当且仅当它对应于$I'$的一个元，因此$D_I$是右$V$ -环当且仅当$I' = I$ ;（d）$D_I$是右\emph{and}左$V$ -环当且仅当$I$是反链;（e）如果$I$有有限的对偶Krull长度，则$D_I$是（f）若$I$至多可数且$I' = \vu$，则$D_I$是可数维$D…

关于弱、s-CS和几乎内射环的若干问题

Aspects on weak, s-CS and almost injective rings

右CF-环（FGF-环）是否是右Artin（拟Frobenius）环尚不清楚。本文在弱CS（s-CS）环和GC ~ 2环的情形下给出了这个问题的肯定回答。在几乎内射环上也得到了一些新的结果。

MacWilliams' extension theorem for infinite rings

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient i…

有界Krull整环的Dubrovin赋值环和素除子的算术伪赋值

Arithmetical pseudo-valuations associated to Dubrovin valuation rings and prime divisors of bounded Krull domains

我们看看在简单的阿廷环和相关的算术伪值Dubrovin赋值环，在诺特的情况下，是$\mathbb{Z}$值的素数的价值函数。这使得有界Krull域的除数理论。

循环是射影和CS或Noether直和的环

Rings Over Which Cyclics are Direct Sums of Projective and CS or Noetherian

称R为右WV -环，如果每个单右R模相对于真圈是内射的.若R是右WV -环，则R是右一致环或右V -环.证明了对右VV-环R，R是右Noether模当且仅当每个右循环模是一个投射模与一个CS或Noether模的直和.对于V -环R上的具有投射脚柱的有限生成模M，使得M的每个子因子都是投射模与CS或Noether模的直和，我们证明了M = X \oplus T，其中X是半单的，T是具有零脚柱的Noether模.在M = R的情形下，得到R = S \oplus T，其中S是半单环，T是具有零脚点的右Noether单环的直和.另外，如果R是von Neumann正则环，则它是半单Artin环.

Artin环上的极小忠实模

Minimal faithful modules over Artinian rings

设R是左Artinian环，M是最小的忠实左R-模，在这个意义下，M没有真子模或真同态像是忠实的.若R是局部的，且socle（R）是R的中心，我们证明了长度（M/J（R）M）+长度（socle（M））$\leq$长度（socle（R））+1，加强了T.古力森其余大部分的文件是专门的情况下，阿廷环R不一定是当地的，并不一定有中央socle。当R是代数闭域上的有限维代数时，我们得到了一个与上述类似的不等式，其中柱脚（R）的长度解释为它作为双模的长度，最后的和数+1由柱脚（R）的结构决定的二部图的欧拉示性数代替.更一般地说，如果不是假设k是代数闭的，而是假设R/J（R）是k上全矩阵代数的直积，并且排除了k具有小有限基数的情况，则该不等式成立。例子表明，对k的基数的限制是必要的，我们不知道我们的结果是否与其他假设显着削弱版本举行。对于只具有一个极小性质的忠实模，没有忠实的真子模或没有忠实的真同态像的情况更简单：在前一种情况下M/J（R）M的长度和在后一种情况下基座（M）的长度是$\leq$长度（基座（R））（再次表示作为双模的长度）。最后一节，基本上独立于其余的笔记，得到这些界限，并表明，每一个忠实模在左Artin环上有一个忠实的子模与前一个极小条件和一个忠实的因子模与后者。证明涉及到模的分解的一些一般性结果。

群体学分级半单环

Groupoid Graded Semisimple Rings

本文研究群体学分级半单环的理论，提出Γ₀-artinian概念，并证明了Wedderburn-Artin定理的群体学分级版本，同时探讨了群体学分级环的自由性及与半单范畴的关系。

M_n（D）的一些极大交换子环

Some Maximal Commutative Subrings of M_n(D)

回答了Weibel的一个问题，证明了Artin半单环的极大交换子环是局部环的直积.

群胚分次环的Hopkins-Levitzki型定理

Hopkins-Levitzki Type Theorems for Groupoid Graded Rings

该研究针对群胚分次环，引入分次理想的左/右逐对象幂零性概念，推广Hopkins-Levitzki定理，证明三类Γ₀-Artinian环为Γ₀-Noetherian环，并用反例说明相关结论。

Morita等价和Morita不变量性质。在Leavitt路代数中的应用

Morita equivalence and Morita invariant properties. Applications in the context of Leavitt path algebras

本文证明了两个幂等元环是Morita等价的，如果其中一个的每个角同构于另一个的矩阵环的角。我们建立了$σ$ -酉环的逆环（一般不成立），其中$σ$ -单位由von Neumann正则元组成.下面的目的是证明一个性质是森田不变的，如果它是不变的下采取角和下采取矩阵。前面的结果被用来检查的森田不变性的某些环的性质（是局部左/右artinian/noether，是范畴左/右artinian，是一个$I_0$ -环，是适当的纯无限）和某些图形的性质的上下文中的莱维特路代数（条件（L），条件（K）和共尾性）。一个不同的证明的事实，即一个图与无限发射器不承认任何desingularization也给出。

Ring Theoretic Properties of Partial Crossed products and related themes

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.

零化理想图具有正亏格的环

Rings Whose Annihilating-Ideal Graphs Have Positive Genus

设$R$是交换环，${\Bbb{A}}(R)$是具有非零化子的理想集. $R$的零化理想图定义为具有点集${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$且两个不同点$I$和$J$相邻的图${\Bbb{AG}}(R)$当且仅当$IJ=(0)$ .研究了零化理想图具有正亏格$γ(\Bbb{AG}(R))$的交换环$R$.证明了若$R$是Artin环使得$γ(\Bbb{AG}(R))<\infty$，则$R$有2个理想或$(R,\mathfrak{m})$是Gorenstein环，其极大理想$\mathfrak{m}$和${\rm v.dim}_{R/{\mathfrak{m}}}{\mathfrak{m}}/{\mathfrak{m}}^{2}=2$ .对于任意两个整数$g\geq 0$和$q>0$，对于$R$的每一个极大理想${\mathfrak{m}}$，都有100多个满足以下条件的Artin环$R$的同构类：（i）$γ(\Bbb{AG}(R)) < g$和（ii）$|R/{\mathfrak{m}}| \leq q$.另外，还证明了若$R$是非整环Noether局部环，使得$γ(\Bbb{AG}(R))<\infty$，则$R$是Gorenstein环或$R$是有2个理想的Artin环.

Wedderburn-Artin 定理

The Wedderburn-Artin Theorem

本文针对 Wedderburn-Artin 定理，提出一种简短、自足且不依赖模论的证明方法。

Graded Frobenius Rings

In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings.

稳定挠理论与单模的内射壳

Stable torsion theories and the injective hulls of simple modules

给出了左诺特环$R$的挠理论刻画，在该环上，单左模的内射壳是局部Artin的.利用挠理论得到了左诺特环满足这一有限性条件的充分条件。

环的最大强左商环

The largest strong left quotient ring of a ring

对任意环$R$，引入了$R$的最大强左商环$Q_l^s(R)$和强左局部化根$\glsR$，并研究了它们的性质.特别地，证明了环$Q_l^s(Q_l^s(R))\simeq Q_l^s(R)$，$\gll^s_{R/\glsR}=0$，并给出了环$ Q_l^s(R)$是半单环的一个判别准则.从经典左商环$Q_{l, cl}(R)$到$Q_l^s(R)$存在一个典型同态，但一般不是同构。给出了几大类环（半素左Goldie环、左Artinian环、具有左Artinian左商环的环等）的对象$Q_l^s(R)$和$\gll^s_R$的显式刻画.

Chain conditions on étale groupoid algebras with applications to Leavitt path algebras and inverse semigroup algebras

The author has previously associated to each commutative ring with unit $R$ and étale groupoid $\mathscr G$ with locally compact, Hausdorff and totally disconnected unit space an $R$-algebra $R\mathscr G$. In this paper we characterize when $R\mathscr G$ is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda~Pino and Siles~Molina of finite dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelma…

Rings which are Essential over their Centers, II

A ring $R$ with center $C$ is said to be\textit{centrally essential} if the module $R_C$ is an essential extension of the module $C_C$. We describe centrally essential exterior algebras of finitely generated free modules over not necessary commutative rings and study properties of semi-Artinian centrally essential rings. V.T.Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A.~ A.A. Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.

关于算术函数环的结构：素理想及其超越

On structures of the ring of arithmetical functions: prime ideals and beyond

本文研究算术函数环的结构性质，证明其既非诺特环也非阿廷环，构造多种素理想并给出非素的半素理想例子，同时证明该环具有无穷克鲁尔维数。

Atiyah猜想与Artinian环

The Atiyah conjecture and Artinian rings

设G是一个有限子群阶有界的群，d表示G的有限子群阶的最低公倍数，K是C的一个复共轭闭子域.设U（G）表示隶属于群von Neumann代数N（G）的无界算子代数，D（KG，U（G））表示KG在U（G）中的除闭包，则D（KG，U（G））是U（G）中包含KG的取逆闭的最小子环.设n为正整数，α \in \Mat_n（KG）.则α诱导出一个有界线性映射α：l^2（G）^\to <$^2（G）^n，并且\ker α有一个定义良好的von Neumann维数\dim_{N(G)}（\ker α）.这是一个非负实数，阿蒂亚猜想的一个版本指出d \dim_{N(G)}（\ker α）\in Z。在这个猜想的假设下，我们将证明：如果G没有非平凡的有限正规子群，则D（KG，U（G））是体上的d \times d矩阵环.我们还将考虑当G有一个非平凡有限正规子群时的情形，以及U（G）的其它包含KG的子环。

The graded structure of algebraic Cuntz-Pimsner rings

The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras. We classify strongly, epsilon-strongly and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. As an application, we characterize noetherian and artinian fractional skew monoid rings by a single corner automorphism.

有限局部主理想环的结构

The structure of finite local principal ideal rings

环$R$称为PIR，如果$R$的每个理想都是主理想。局部环$(R,\mf{m)}$是Artin PIR当且仅当它的极大理想$\mf{m}$是主环且有有限幂零指数.本文确定了有限局部PIR的结构。

Weakly curved A-infinity algebras over a topological local ring

We define and study the derived categories of the first kind for curved DG and A-infinity algebras complete over a pro-Artinian local ring with the curvature elements divisible by the maximal ideal of the local ring. We develop the Koszul duality theory in this setting and deduce the generalizations of the conventional results about A-infinity modules to the weakly curved case. The formalism of contramodules and comodules over pro-Artinian topological rings is used throughout the paper. Our motivation comes from the Floer-Fukaya theory.

Connections between Representation-Finite and Köthe Rings

A ring $R$ is called left $k$-cyclic if every left $R$-module is a direct sum of indecomposable modules which are homomorphic image of $_{R}R^k$. In this paper, we give a characterization of left $k$-cyclic rings. As a consequence, we give a characterization of left Köthe rings, which is a generalization of Köthe-Cohen-Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left $k$-cyclic ring. As a corollary, we show that $R$ is Morita equivalent to a basic left Köthe ring if and only if $R$ is an artinian left multiplicity-free top ring.