• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.371-381
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• 2020

A ring R is called right pure-injective if it is injective with respect to pure exact sequences. According to a well known result of L. Melkersson, every commutative Artinian ring is pure-injective, but the converse is not true, even if R is a commutative Noetherian local ring. In this paper, a series of conditions under which right pure-injective rings are either right Artinian rings or quasi-Frobenius rings are given. Also, some of our results extend previously known results for quasi-Frobenius rings.

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.711-719
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• 2018

In this paper we define and study a new kind of dimension called, semisimple dimension, that measures how far a module is from being semisimple. Like other kinds of dimensions, this is an ordinal valued invariant. We give some interesting and useful properties of rings or modules which have semisimple dimension. It is shown that a noetherian module with semisimple dimension is an artinian module. A domain with semisimple dimension is a division ring. Also, for a semiprime right non-singular ring R, if its maximal right quotient ring has semisimple dimension as a right R-module, then R is a semisimple artinian ring. We also characterize rings whose modules have semisimple dimension. In fact, it is shown that all right R-modules have semisimple dimension if and only if the free right R-module ${\oplus}^{\infty}_{i=1}$ R has semisimple dimension, if and only if R is a semisimple artinian ring.

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.457-470
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• 2005

Yu showed that every right (left) primitive factor ring of weakly right (left) duo rings is a division ring. It is not difficult to show that each weakly right (left) duo ring is abelian and has the classical right (left) quotient ring. In this note we first provide a left duo ring (but not weakly right duo) in spite of it being left Noetherian and local. Thus we observe conditions under which weakly one-sided duo rings may be two-sided. We prove that a weakly one-sided duo ring R is weakly duo under each of the following conditions: (1) R is semilocal with nil Jacobson radical; (2) R is locally finite. Based on the preceding case (1) we study a kind of composition length of a right or left Artinian weakly duo ring R, obtaining that i(R) is finite and $\alpha^{i(R)}R\;=\;R\alpha^{i(R)\;=\;R\alpha^{i(R)}R\;for\;all\;\alpha\;{\in}\;R$, where i(R) is the index (of nilpotency) of R. Note that one-sided Artinian rings and locally finite rings are strongly $\pi-regular$. Thus we also observe connections between strongly $\pi-regular$ weakly right duo rings and related rings, constructing available examples.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.45-51
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• 2018

This paper contains some results that grew out of an attempt to Camillo-Krause conjecture: Is a ring R right Noetherian if for each nonzero right ideal I of R, R/I is an Artinian right R-module?

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1711-1723
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• 2013

P. V$\acute{a}$mos called a ring R 2-good if every element is the sum of two units. The ring of all $n{\times}n$ matrices over an elementary divisor ring is 2-good. A (right) self-injective von Neumann regular ring is 2-good provided it has no 2-torsion. Some of the earlier results known to us about 2-good rings (although nobody so called at those times) were due to Ehrlich, Henriksen, Fisher, Snider, Rapharl and Badawi. We continue in this paper the study of 2-good rings by several authors. We give some examples of 2-good rings and their related properties. In particular, it is shown that if R is an exchange ring with Artinian primitive factors and 2 is a unit in R, then R is 2-good. We also investigate various kinds of extensions of 2-good rings, including the polynomial extension, Nagata extension and Dorroh extension.

• The Pure and Applied Mathematics
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• v.1 no.1
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• pp.1-6
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• 1994

In this paper, some properties for a PI-ring satisfying the descending chain condition on essential left ideals are studied: Let R be a ring with a polynomial identity satisfying the descending chain condition on essential ideals. Then all minimal prime ideals in R are maximal ideals. Moreover, if R has only finitely many minimal prime ideals, then R is left and right Artinian. Consequently, if every primeideal of R is finitely generated as a left ideal, then R is left and right Artinian. A finitely generated PI-algebra over a commutative Noetherian ring satisfying the descending chain condition on essential left ideals is a finite module over its center.(omitted)

• East Asian mathematical journal
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• v.6 no.2
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• pp.167-182
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• 1990

The relationships between submodules of a module and ideals of the endomorphism ring of a module had been studied in [1]. For a submodule L of a moudle M, the set $I^L$ of all endomorphisms whose images are contained in L is a left ideal of the endomorphism ring End (M) and for a submodule N of M, the set $I_N$ of all endomorphisms whose kernels contain N is a right ideal of End (M). In this paper, author defines an H-invariant module and proves that every submodule of an H-invariant module is the image and kernel of unique endomorphisms. Every ideal $I^L(I_N)$ of the endomorphism ring End(M) when M is H-invariant is a left (respectively, right) principal ideal of End(M). From the above results, if a module M is H-invariant then each left, right, or both sided ideal I of End(M) is an intersection of a left, right, or both sided principal ideal and I itself appropriately. If M is an H-invariant module then the ACC on the set of all left ideals of type $I^L$ implies the ACC on M. Also if the set of all right ideals of type $I^L$ has DCC, then H-invariant module M satisfies ACC. If the set of all left ideals of type $I^L$ satisfies DCC, then H-invariant module M satisfies DCC. If the set of all right ideals of type $I_N$ satisfies ACC then H-invariant module M satisfies DCC. Therefore for an H-invariant module M, if the endomorphism ring End(M) is left Noetherian, then M satisfies ACC. And if End(M) is right Noetherian then M satisfies DCC. For an H-invariant module M, if End(M) is left Artinian then M satisfies DCC. Also if End(M) is right Artinian then M satisfies ACC.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.655-663
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• 2014

Let R be a ring with identity, X(R) the set of all nonzero, non-units of R and G(R) the group of all units of R. We show that for a matrix ring $M_n(D)$, $n{\geq}2$, if a, b are singular matrices of the same rank, then ${\mid}o_{\ell}(a){\mid}={\mid}o_{\ell}(b){\mid}$, where $o_{\ell}(a)$ and $o_{\ell}(b)$ are the orbits of a and b, respectively, under the left regular action. We also show that for a semisimple Artinian ring R such that $X(R){\neq}{\emptyset}$, $$R{{\sim_=}}{\oplus}^m_{i=1}M_n_i(D_i)$$, with $D_i$ infinite division rings of the same cardinalities or R is isomorphic to the ring of $2{\times}2$ matrices over a finite field if and only if ${\mid}o_{\ell}(x){\mid}={\mid}o_{\ell}(y){\mid}$ for all $x,y{\in}X(R)$.

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.277-285
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• 2017

In the various module generalizations of the concepts of prime (semiprime) for a ring, the question "when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If R is Morita equivalent to a right pre-duo ring (e.g., if R is commutative) then weakly compressible R-modules are precisely subdirect products of prime R-modules if and only if dim(R) = 0 and R/N(R) is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible R-modules is an enveloping for Mod-R. Some related conditions are also investigated.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.25-40
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• 2018

We focus on the structure of the set of noncentral idempotents whose role is similar to one of central idempotents. We introduce the concept of quasi-Abelian rings which unit-regular rings satisfy. We first observe that the class of quasi-Abelian rings is seated between Abelian and direct finiteness. It is proved that a regular ring is directly finite if and only if it is quasi-Abelian. It is also shown that quasi-Abelian property is not left-right symmetric, but left-right symmetric when a given ring has an involution. Quasi-Abelian property is shown to do not pass to polynomial rings, comparing with Abelian property passing to polynomial rings.