• 대한수학회보
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• 제57권2호
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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.

• 대한수학회보
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• 제43권4호
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• pp.755-762
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• 2006

For a commutative ring R and an injective cogenerator E in the category of R-modules, we characterize von Neumann regular rings and semisimple artinian rings in terms of the properties of dual modules with respect to E.

• 대한수학회보
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• 제39권1호
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• pp.1-8
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• 2002

We characterize the commutative Artinian rings R every proper quotient ring (respectively every proper ideal) in which is invariant with respect to all derivations.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제7권1호
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• pp.33-40
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• 2000

We prove some characterization of rings with chain conditions in terms of fuzzy quotient rings and fuzzy ideals. We also show that a ring R is left Artinian if and only of the set of values of every fuzzy ideal on R is upper well-ordered.

• 대한수학회보
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• 제31권1호
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• pp.35-44
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• 1994

In this paper we study exponential growth of Betti numbers over artinian local rings. By the Change of Tor Formula the results in the paper extend to the asymptotic behavior of Betti numbers over Cohen-Macaulay local rings. Using the length function of an artinian ring we calculate an upper bound for the number of generators of modules, this is then used to maximize the number of generators of sygyzy modules. Finally, applying a filtration of an ideal, which we call a Loewy series of an ideal, we derive an invariant B(R) of an artinian local ring R, such that if B(R)>1, then the sequence $b^{R}$$_{i}$ (M) of Betti numbers is strictly increasing and has strong exponential growth for any finitely generated non-free R-module M (Theorem 2.7).).

• 대한수학회논문집
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• 제10권4호
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• pp.839-847
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• 1995

For the given torsion theory $\tau$, we study some equivalent conditions when the localized ring $R_\tau$ be semisimple artinian (Theorem 4). Using this, if $R_\tau$ is semisimple artinian ring, we study when does the given ring R become left V-ring?

• 대한수학회보
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• 제59권5호
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• pp.1317-1325
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• 2022

In this note, we prove that an Artinian local ring is G-semisimple (resp., SG-semisimple, 2-SG-semisimple) if and only if its maximal ideal is G-projective (resp., SG-projective, 2-SG-projective). As a corollary, we obtain the global statement of the above. We also give some examples of local G-semisimple rings whose maximal ideals are n-generated for some positive integer n.

• Kyungpook Mathematical Journal
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• 제62권3호
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• pp.407-423
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• 2022

In this paper, we determine the structure of rings that have secondary representation (called representable rings) and give some characterizations of these rings. Also, we characterize Artinian rings in terms of representable rings. Then we introduce completely representable modules, modules every non-zero submodule of which have secondary representation, and give some necessary and sufficient conditions for a module to be completely representable. Finally, we define strongly representable modules and give some conditions under which representable module is strongly representable.

• 대한수학회지
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• 제42권3호
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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.

• 대한수학회보
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• 제29권2호
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• pp.221-226
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• 1992

The involutions in a left Artinian ring A with identity are investigated. Those left Artinian rings A for which 2 is a unit in A and the set of involutions in A forms a finite abelian group are characterized by the number of involutions in A.