• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1123-1139
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• 2017

The aim in this note is to describe some classes of rings in relation to factorization by prime radical, upper nilradical, and Jacobson radical. We introduce the concepts of tpr ring, tunr ring, and tjr ring in the process, respectively. Their ring theoretical structures are investigated in relation to various sorts of factor rings and extensions. We also study the structure of noncommutative tpr (tunr, tjr) rings of minimal order, which can be a base of constructing examples of various ring structures. Various sorts of structures of known examples are studied in relation with the topics of this note.

• Korean Journal of Mathematics
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• v.28 no.1
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• pp.65-73
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• 2020

This article concerns a property of local rings and domains. A ring R is called weakly local if for every a ∈ R, a is regular or 1-a is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1019-1028
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• 2023

Let 𝔄 and 𝔅 be unital prime *-algebras such that 𝔄 contains a nontrivial projection. In the present paper, we show that if a bijective map Θ : 𝔄 → 𝔅 satisfies Θ(_*[X ⋄ Y, Z]) = _*[Θ(X) ⋄ Θ(Y), Θ(Z)] for all X, Y, Z ∈ 𝔄, then Θ or -Θ is a *-ring isomorphism. As an application, we shall characterize such maps in factor von Neumann algebras.

• Journal of Chest Surgery
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• v.38 no.2 s.247
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• pp.132-138
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• 2005

Background: Mitral valve repair (MVP) is the optimal procedure for mitral regurgitation (MR), however, failure and subsequent reoperations are the limitations. The current study assessed the procedure in relation to the primary valve related causes of recurrent MR. Material and Method: MR was treated in 493 patients undergoing MVP from January of 1994 to January of 2002. The causes of MR were degenerative $(n=252,\;51.5\%),$ rheumatic $(n=156,\; 31.6\%),$ and others $(n=85,\; 16.9\%).$ Surgery comprised 446 ring annuloplasties $(90.5\%),$ 227 new chordae formations $(46\%),$ 125 quadriangular resections $(25.3\%),$ 28 chordae transfers $(5.7\%),$ and 8 Alfieri's stitches $(1.6\%).$ The mean follow up was $29.04\pm22.81$ months. Result: There were 5 early $(1.01\%)$, and 5 late deaths $(1.01\%).$ The reoperation rate was $1.42\%$. There were 45 $(9.1\%)$ recurrent MR (grade III or IV). Of these, 24 were procedure related including incomplete repair (n=14), discordant new chordae length (n=8) and others (n=2). In 21 patients, the cause was valve related including rheumatic disease progression (n=10), recurrent chordae elongation or prolapse (n=5) and others (n=6). Severe MR was higher after incomplete repair (p < 0.001), and valve related failure strongly correlated with rheumatic progression (p < 0.05). Conclusion: Since completeness of operation is the prime risk factor that determine the repair durability, intra-operative assessment of the initial repair with trans-esophageal echocardiography is essential.