• Korean Journal of Mathematics
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• v.8 no.1
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• pp.87-90
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• 2000

Let R be an integral domain with identity. We show that each associated prime ideal of a principal ideal in R[X] has height one if and only if each associated prime ideal of a principal ideal in R has height one and R is an S-domain.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.595-613
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• 2022

Let R be a commutative ring with identity and let I be a proper ideal of R. In this paper, we study the ideal based extended zero-divisor graph 𝚪'_I (R) and prove that 𝚪'_I (R) is connected with diameter at most two and if 𝚪'_I (R) contains a cycle, then girth is at most four girth at most four. Furthermore, we study affinity the connection between the ideal based extended zero-divisor graph 𝚪'_I (R) and the ideal-based zero-divisor graph 𝚪_I (R) associated with the ideal I of R. Among the other things, for a radical ideal of a ring R, we show that the ideal-based extended zero-divisor graph 𝚪'_I (R) is identical to the ideal-based zero-divisor graph 𝚪_I (R) if and only if R has exactly two minimal prime-ideals which contain I.

• Kyungpook Mathematical Journal
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• v.58 no.3
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• pp.481-488
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• 2018

Let I be a ${\ast}$-ideal on a 2-torsion free ${\ast}$-prime ring and $F:R{\rightarrow}R$ a generalized derivation with an associated derivation $d:R{\rightarrow}R$. The aim of this paper is to explore the condition under which generalized derivation F becomes a left centralizer i.e., associated derivation d becomes a trivial map (i.e., zero map) on R.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.539-548
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• 2024

Let R be a ring and P be a prime ideal of R. A mapping d : R → R is called a multiplicative derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R. In this paper, our main motive is to obtain the well-known theorem due to Posner in the ring R/P for generalized derivations associated with a multiplicative derivation defined by an additive mapping F : R → R such that F(xy) = F(x)y + xd(y), where d : R → R is a multiplicative derivation not necessarily additive. This article discusses the use of generalized derivations associated with a multiplicative derivation to investigate the commutativity of the quotient ring R/P.

• Honam Mathematical Journal
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• v.39 no.2
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• pp.275-296
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• 2017

All rings considered here are commutative rings with identity and all modules considered here are unital left modules. A submodule N of an R-module M is said to be extended to M if $N=aM$ for some ideal a of R and it is said to be fully invariant if ${\varphi}(L){\subseteq}L$ for every ${\varphi}{\in}End(M)$. An R-module M is called a [resp., fully invariant] multiplication module if every [resp., fully invariant] submodule is extended to M. The class of fully invariant multiplication modules is bigger than the class of multiplication modules. We deal with prime submodules and associated prime submodules of fully invariant multiplication modules. In particular, when M is a nonzero faithful multiplication module over a Noetherian ring, we characterize the zero-divisors of M in terms of the associated prime submodules, and we show that the set Aps(M) of associated prime submodules of M determines the set $Zdv_M(M)$ of zero-dvisors of M and the support Supp(M) of M.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.101-106
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• 2006

Let R be a prime ring and I a nonzero ideal of R. Let $\alpha,\;\nu,\;\tau\;R{\rightarrow}R$ be the endomorphisms and $\beta,\;\mu\;R{\rightarrow}R$ the automorphisms. If R admits a generalized $(\alpha,\;\beta)-derivation$ g associated with a nonzero $(\alpha,\;\beta)-derivation\;\delta$ such that $g([\mu(x),y])\;=\;[\nu/(x),y]\alpha,\;\tau$ for all x, y ${\in}I$, then R is commutative.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.495-502
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• 2017

Let R be a 2-torsion free prime ring and J be a nonzero Jordan ideal of R. Let F and G be two generalized derivations with associated derivations f and g, respectively. Our main result in this paper shows that if F(x)x - xG(x) = 0 for all $x{\in}J$, then R is commutative and F = G or G is a left multiplier and F = G + f. This result with its consequences generalize some recent results due to El-Soufi and Aboubakr in which they assumed that the Jordan ideal J is also a subring of R.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.69-75
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• 1997

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.511-528
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• 2008

Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and $\upsilon$ be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple $\upsilon$-ideals $m\;=\;P_0\;{\supset}\;P_1\;{\supset}\;{\cdots}\;{\supset}\;P_t\;=\;P$ and all the other $\upsilon$-ideals are uniquely factored into a product of those simple ones [17]. Lipman further showed that the predecessor of the smallest simple $\upsilon$-ideal P is either simple or the product of two simple $\upsilon$-ideals. The simple integrally closed ideal P is said to be free for the former and satellite for the later. In this paper we describe the sequence of simple $\upsilon$-ideals when P is satellite of order 3 in terms of the invariant $b_{\upsilon}\;=\;|\upsilon(x)\;-\;\upsilon(y)|$, where $\upsilon$ is the prime divisor associated to P and m = (x, y). Denote $b_{\upsilon}$ by b and let b = 3k + 1 for k = 0, 1, 2. Let $n_i$ be the number of nonmaximal simple $\upsilon$-ideals of order i for i = 1, 2, 3. We show that the numbers $n_{\upsilon}$ = ($n_1$, $n_2$, $n_3$) = (${\lceil}\frac{b+1}{3}{\rceil}$, 1, 1) and that the rank of P is ${\lceil}\frac{b+7}{3}{\rceil}$ = k + 3. We then describe all the $\upsilon$-ideals from m to P as products of those simple $\upsilon$-ideals. In particular, we find the conductor ideal and the $\upsilon$-predecessor of the given ideal P in cases of b = 1, 2 and for b = 3k + 1, 3k + 2, 3k for $k\;{\geq}\;1$. We also find the value semigroup $\upsilon(R)$ of a satellite simple valuation ideal P of order 3 in terms of $b_{\upsilon}$.

• Kyungpook Mathematical Journal
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• v.58 no.1
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• pp.9-17
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• 2018

Let $R ={\oplus}_{n{\in}Z}R_n$ be a graded commutative Noetherian ring and let I be a graded ideal of R and J be an arbitrary ideal. It is shown that the i-th generalized local cohomology module of graded module M with respect to the (I, J), is graded. Also, the asymptotic behaviour of the homogeneous components of $H^i_{I,J}(M)$ is investigated for some i's with a specified property.