• 대한수학회보
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• 제54권4호
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• pp.1281-1291
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• 2017

Let R be a strongly 2-primal ring and I a proper ideal of R. Then there are only finitely many prime ideals minimal over I if and only if for every prime ideal P minimal over I, the ideal $P/{\sqrt{I}}$ of $R/{\sqrt{I}}$ is finitely generated if and only if the ring $R/{\sqrt{I}}$ satisfies the ACC on right annihilators. This result extends "D. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math. Soc. 122 (1994), no. 1, 13-14." to large classes of noncommutative rings. It is also shown that, a 2-primal ring R only has finitely many minimal prime ideals if each minimal prime ideal of R is finitely generated. Examples are provided to illustrate our results.

• 대한수학회보
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• 제42권3호
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• pp.477-484
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• 2005

Let R be a ring with an automorphism 17. An ideal [ of R is ($\sigma$-ideal of R if $\sigma$(I).= I. A proper ideal P of R is ($\sigma$-prime ideal of R if P is a $\sigma$-ideal of R and for $\sigma$-ideals I and J of R, IJ $\subseteq$ P implies that I $\subseteq$ P or J $\subseteq$ P. A proper ideal Q of R is $\sigma$-semiprime ideal of Q if Q is a $\sigma$-ideal and for a $\sigma$-ideal I of R, I$^{2}$ $\subseteq$ Q implies that I $\subseteq$ Q. The $\sigma$-prime radical is defined by the intersection of all $\sigma$-prime ideals of R and is denoted by P$_{(R). In this paper, the following results are obtained: (1) For a principal ideal domain R, P$_{(R) is the smallest $\sigma$-semiprime ideal of R; (2) For any ring R with an automorphism $\sigma$ and for a skew Laurent polynomial ring R[x, x$^{-1}$; $\sigma$], the prime radical of R[x, x$^{-1}$; $\sigma$] is equal to P$_{(R)[x, x$^{-1}$; $\sigma$ ].

• East Asian mathematical journal
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• 제18권2호
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• pp.271-282
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• 2002

Let R be a ring with an endomorphism $\sigma$ and a derivation $\delta$. An ideal I of R is ($\sigma,\;\delta$)-ideal of R if $\sigma(I){\subseteq}I$ and $\delta(I){\subseteq}I$. An ideal P of R is a ($\sigma,\;\delta$)-prime ideal of R if P(${\neq}R$) is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideals I and J of R, $IJ{\subseteq}P$ implies that $I{\subseteq}P$ or $J{\subseteq}P$. An ideal Q of R is ($\sigma,\;\delta$)-semiprime ideal of R if Q is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideal I of R, $I^2{\subseteq}Q$ implies that $I{\subseteq}Q$. The ($\sigma,\;\delta$)-prime radical (resp. prime radical) is defined by the intersection of all ($\sigma,\;\delta$)-prime ideals (resp. prime ideals) of R and is denoted by $P_{(\sigma,\delta)}(R)$(resp. P(R)). In this paper, the following results are obtained: (1) $P_{(\sigma,\delta)}(R)$ is the smallest ($\sigma,\;\delta$)-semiprime ideal of R; (2) For every extended endomorphism $\bar{\sigma}$ of $\sigma$, the $\bar{\sigma}$-prime radical of an Ore extension $P(R[x;\sigma,\delta])$ is equal to $P_{\sigma,\delta}(R)[x;\sigma,\delta]$.

• 대한수학회보
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• 제57권1호
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• pp.117-126
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• 2020

Let R be a commutative ring with identity. A proper ideal I of R is called 2-prime if for all a, b ∈ R such that ab ∈ I, then either a² or b² lies in I. In this paper, we study 2-prime ideals which are generalization of prime ideals. Our study provides an analogous to the prime avoidance theorem and some applications of this theorem. Also, it is shown that if R is a PID, then the families of primary ideals and 2-prime ideals of R are identical. Moreover, a number of examples concerning 2-prime ideals are given. Finally, rings in which every 2-prime ideal is a prime ideal are investigated.

• 대한수학회보
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• 제20권1호
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• pp.45-49
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• 1983

If R is ring and M is a right (or left) R-module, then M is called a faithful R-module if, for some a in R, x.a=0 for all x.mem.M then a=0. In [4], R.E. Johnson defines that M is a prime module if every non-zero submodule of M is faithful. Let us define that M is of prime type provided that M is faithful if and only if every non-zero submodule is faithful. We call a right (left) ideal I of R is of prime type if R/I is of prime type as a R-module. This is equivalent to the condition that if xRy.subeq.I then either x.mem.I ro y.mem.I (see [5:3:1]). It is easy to see that in case R is a commutative ring then a right or left ideal of a prime type is just a prime ideal. We have defined in [5], that a chain of right ideals of prime type in a ring R is a finite strictly increasing sequence I$_{0}$.contnd.I$_{1}$.contnd....contnd.I$_{n}$; the length of the chain is n. By the right dimension of a ring R, which is denoted by dim, R, we mean the supremum of the length of all chains of right ideals of prime type in R. It is an integer .geq.0 or .inf.. The left dimension of R, which is denoted by dim$_{l}$ R is similarly defined. It was shown in [5], that dim$_{r}$R=0 if and only if dim$_{l}$ R=0 if and only if R modulo the prime radical is a strongly regular ring. By "a strongly regular ring", we mean that for every a in R there is x in R such that axa=a=a$^{2}$x. It was also shown that R is a simple ring if and only if every right ideal is of prime type if and only if every left ideal is of prime type. In case, R is a (right or left) primitive ring then dim$_{r}$R=n if and only if dim$_{l}$ R=n if and only if R.iden.D$_{n+1}$ , n+1 by n+1 matrix ring on a division ring D. in this paper, we establish the following results: (1) If R is prime ring and dim$_{r}$R=n then either R is a righe Ore domain such that every non-zero right ideal of a prime type contains a non-zero minimal prime ideal or the classical ring of ritght quotients is isomorphic to m*m matrix ring over a division ring where m.leq.n+1. (b) If R is prime ring and dim$_{r}$R=n then dim$_{l}$ R=n if dim$_{l}$ R=n if dim$_{l}$ R<.inf. (c) Let R be a principal right and left ideal domain. If dim$_{r}$R=1 then R is an unique factorization domain.TEX>R=1 then R is an unique factorization domain.

• 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).

• East Asian mathematical journal
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• 제22권2호
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• pp.239-248
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• 2006

Let G be a group acting on a ring R. We will define the group action of G on an intuitionsitic fuzzy set of R. We will introduce intuitionistic fuzzy G-prime ideals of a ring and we will prove that every intuitionistic fuzzy G-prime ideal is the largest G-invariant intuitionistic fuzzy ideal of R contained in the intuitionistic fuzzy prime ideal which is uniquely determined up to G-orbits.

• 대한수학회보
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• 제48권2호
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• pp.277-290
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• 2011

We show that the ${\theta}$-prime radical of a ring R is the set of all strongly ${\theta}$-nilpotent elements in R, where ${\theta}$ is an automorphism of R. We observe some conditions under which the ${\theta}$-prime radical of coincides with the prime radical of R. Moreover we characterize elements in prime radicals of skew Laurent polynomial rings, studying (${\theta}$, ${\theta}^{-1}$)-(semi)primeness of ideals of R.

• East Asian mathematical journal
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• 제21권1호
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• pp.1-7
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• 2005

The object of this paper is to study($\sigma,\;\tau$)-Lie ideals in semi-prime rings with derivation. Main result is the following theorem: Let R be a semi-prime ring with 2-torsion free, $\sigma$ and $\tau$ two automorphisms of R such that $\sigma\tau=\tau\sigma$=, U be both a non-zero ($\sigma,\;\tau$)-Lie ideal and subring of R. If $d^2(U)=0$, then d(U)=0 where d a non-zero derivation of R such that $d\sigma={\sigma}d,\;d\tau={\tau}d$.

• 대한수학회보
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• 제58권5호
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• pp.1069-1078
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• 2021

Let R be a commutative ring with identity. A proper ideal I of R is called 1-absorbing primary ([4]) if for all nonunit a, b, c ∈ R such that abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. The concept of 1-absorbing primary ideals in a polynomial ring, in a PID and in idealization of a module is studied. Moreover, we introduce weakly 1-absorbing primary ideals which are generalization of weakly prime ideals and 1-absorbing primary ideals. A proper ideal I of R is called weakly 1-absorbing primary if for all nonunit a, b, c ∈ R such that 0 ≠ abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. Some properties of weakly 1-absorbing primary ideals are investigated. For instance, weakly 1-absorbing primary ideals in decomposable rings are characterized. Among other things, it is proved that if I is a weakly 1-absorbing primary ideal of a ring R and 0 ≠ I₁I₂I₃ ⊆ I for some ideals I₁, I₂, I₃ of R such that I is free triple-zero with respect to I₁I₂I₃, then I₁I₂ ⊆ I or I₃ ⊆ I.