• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1177-1193
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• 2020

Fully prime rings (in which every proper ideal is prime) have been studied by Blair and Tsutsui, and fully semiprime rings (in which every proper ideal is semiprime) have been studied by Courter. For a given module M, we introduce the notions of a fully prime module and a fully semiprime module, and extend certain results of Blair, Tsutsui, and Courter to the category subgenerated by M. We also consider the relationship between the conditions (1) M is a fully prime (semiprime) module, and (2) the endomorphism ring of M is a fully prime (semiprime) ring.

• 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.

• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.615-620
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• 2003

Semi-meet-prime projective modules and fully invariant meet-prime submodules of a projective module are studied. Actually a generalization of the Schur's lemma and semi-prime endmorphism rings of projective modules are considered.

• Communications of the Korean Mathematical Society
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• v.19 no.4
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• pp.619-637
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• 2004

In this paper, a left module over an associative ring with identity is defined to be openly semiprimitive (strongly semiprimitive, respectively) by the zero intersection of all maximal open fully invariant submodules (all maximal open submodules which are fully invariant, respectively) of it. For any projective module, the openly semiprimitivity of the projective module is an equivalent condition of the semiprimitivity of endomorphism ring of the projective module and the strongly semiprimitivity of the projective module is an equivalent condition of the endomorphism ring of the projective module being a sub direct product of a set of subdivisions of division rings.

• The Bulletin of The Korean Astronomical Society
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• v.39 no.2
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• pp.90-90
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• 2014

The Immersion Grating Infrared Spectrometer (IGRINS) is the first astronomical spectrograph that uses a silicon immersion grating as its dispersive element. IGRINS fully covers the H and K band atmospheric transmission windows in a single exposure. It is a compact high-resolution cross-dispersion spectrometer whose resolving power R is 40,000. An individual volume phase holographic grating serves as a secondary dispersing element for each of the H and K spectrograph arms. On the 2.7m Harlan J. Smith telescope at the McDonald Observatory, the slit size is $1^{{\prime}{\prime}}{\times}15^{{\prime}{\prime}}$. IGRINS has a plate scale of 0.27" pixel-1 on a $2048{\times}2048$ pixel Teledyne Scientific & Imaging HAWAII-2RG detector with a SIDECAR ASIC cryogenic controller. The instrument includes four subsystems; a calibration unit, an input relay optics module, a slit-viewing camera, and nearly identical H and K spectrograph modules. The use of a silicon immersion grating and a compact white pupil design allows the spectrograph collimated beam size to be 25mm, which permits the entire cryogenic system to be contained in a moderately sized ($0.96m{\times}0.6m{\times}0.38m$) rectangular Dewar. The fabrication and assembly of the optical and mechanical components were completed in 2013. From January to July of this year, we completed the system optical alignment and carried out commissioning observations on three runs to improve the efficiency of the instrument software and hardware. We describe the major design characteristics of the instrument including the system requirements and the technical strategy to meet them. We also present the instrumental performance test results derived from the commissioning runs at the McDonald Observatory.