• 대한수학회보
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• 제50권1호
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• pp.11-24
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• 2013

Let m and n be fixed positive integers and M a right R-module. Recall that M is said to be ($m$, $n$)-injective if $Ext^1$(P, M) = 0 for any ($m$, $n$)-presented right R-module P; M is said to be ($m$, $n$)-flat if $Tor_1$(N, P) = 0 for any ($m$, $n$)-presented left R-module P. In terms of some derived functors, relative injective or relative flat resolutions and dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and p.p. rings are given.

• 대한수학회보
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• 제58권4호
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• pp.897-908
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• 2021

In this paper, we introduce the notion of pseudo 2-prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity. A proper ideal P of R is said to be a pseudo 2-prime ideal if whenever xy ∈ P for some x, y ∈ R, then x²ⁿ ∈ Pⁿ or y²ⁿ ∈ Pⁿ for some n ∈ ℕ. Various examples and properties of pseudo 2-prime ideals are given. We also characterize pseudo 2-prime ideals of PID's and von Neumann regular rings. Finally, we use pseudo 2-prime ideals to characterize almost valuation domains (AV-domains).

• 대한수학회보
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• 제56권3호
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• pp.729-743
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• 2019

In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let R be a commutative ring with nonzero identity and Q a proper ideal of R. Then Q is called strongly quasi primary if $ab{\in}Q$ for $a,b{\in}R$ implies either $a^2{\in}Q$ or $b^n{\in}Q$ ($a^n{\in}Q$ or $b^2{\in}Q$) for some $n{\in}{\mathbb{N}}$. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph ${\Gamma}_I(R)$ and denote it by ${\Gamma}^*_I(R)$, where I is an ideal of R. We investigate the relations between ${\Gamma}^*_I(R)$ and ${\Gamma}_I(R)$. Further, we use strongly quasi primary ideals and ${\Gamma}^*_I(R)$ to characterize von Neumann regular rings.