• 대한수학회보
• /
• 제51권1호
• /
• pp.253-266
• /
• 2014

Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.

• 대한수학회논문집
• /
• 제38권1호
• /
• pp.1-9
• /
• 2023

An le-module M over a commutative ring R is a complete lattice ordered additive monoid (M, ⩽, +) having the greatest element e together with a module like action of R. This article characterizes the le-modules _RM such that the pseudo-prime spectrum X_M endowed with the Zariski topology is a Noetherian topological space. If the ring R is Noetherian and the pseudo-prime radical of every submodule elements of _RM coincides with its Zariski radical, then X_M is a Noetherian topological space. Also we prove that if R is Noetherian and for every submodule element n of M there is an ideal I of R such that V (n) = V (Ie), then the topological space X_M is spectral.

• 대한수학회지
• /
• 제34권1호
• /
• pp.69-75
• /
• 1997

• East Asian mathematical journal
• /
• 제12권1호
• /
• pp.1-5
• /
• 1996

• 호남수학학술지
• /
• 제3권1호
• /
• pp.19-21
• /
• 1981

In this note, with respect to the quotient ring, we give a property of those rings in which every prime ideal is maximal.

• 한국지능시스템학회논문지
• /
• 제9권5호
• /
• pp.509-512
• /
• 1999

We study semiprime fuzzy ideals semiprimary fuzzy ideals and their properties. We investigate that if a fuzzy ideal is semiprime and semiprimary then it is prime.

• 대한수학회보
• /
• 제38권3호
• /
• pp.543-549
• /
• 2001

We show that R[X] is a Krull (Resp. factorial) ring if and only if R is a normal Krull (resp, factorial) ring with a finite number of minimal prime ideals if and only if R is a Krull (resp. factorial) ring with a finite number of minimal prime ideals and R(sub)M is an integral domain for every maximal ideal M of R. As a corollary, we have that if R[X] is a Krull (resp. factorial) ring and if D is a Krull (resp. factorial) overring of R, then D[X] is a Krull (resp. factorial) ring.

• 대한수학회논문집
• /
• 제28권4호
• /
• pp.669-677
• /
• 2013

In this paper we introduce the notion of NI near-rings similar to the notion introduced in rings. We give topological properties of collection of strongly prime ideals in NI near-rings. We have shown that if N is a NI and weakly pm near-ring, then $Max(N)$ is a compact Hausdorff space. We have also shown that if N is a NI near-ring, then for every $a{\in}N$, $cl(D(a))=V(N^*(N)_a)=Supp(a)=SSpec(N){\setminus}int\;V(a)$.

• Kyungpook Mathematical Journal
• /
• 제63권3호
• /
• pp.325-331
• /
• 2023

Let 𝔄 be a prime ring of char(𝔄 ≠ 2, ℒ a non-central Lie ideal of 𝔄, ℱ a generalized skew derivation of 𝔄 and p ∈ 𝔄, a nonzero fixed element. If pℱ(η)η ∈ C for any η ∈ ℒ, then 𝔄 satisfies S₄.

• 대한수학회논문집
• /
• 제39권3호
• /
• pp.595-609
• /
• 2024

In this article, we study a certain class of partitioning ideals known as Q-ideals, in semirings. Main objective is to investigate differential identities linking a semiring S to its prime Q-ideal I_Q, which ensure the commutativity and other features of S/I_Q.