• Kyungpook Mathematical Journal
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• v.52 no.2
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• pp.179-188
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• 2012

We investigate in this paper rings containing a non-essential $nil$-injective maximal left ideal. We show that if R is a left MC2 ring containing a non-essential $nil$-injective maximal left ideal, then R is a left $nil$-injective ring. Using this result, some known results are extended.

• Journal of applied mathematics & informatics
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• v.41 no.6
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• pp.1173-1180
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• 2023

In this paper, rings in which essential maximal right ideals are GP-injective are studied. Whether the rings with this condition satisfy von Neumann regularity is the goal of this study. The obtained research results are twofold: First, it was shown that this regularity holds even when the reduced ring is replaced with π-IFP and NI-ring. Second, it was shown that this regularity also holds even when the maximal right ideal is changed to GW-ideal. This can be interpreted as an extension of the existing results.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.653-659
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• 1995

We prove that two essentially normal elements of a type $II_{\infty}$ factor von Neumann algebra are unitarily equivalent up to the compact ideal if and only if they have the identical essential spectrum and the same index data. Also we calculate the spectrum and essential spectrum of a non-unitary isometry of von Neumann algebra.

• The Mathematical Education
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• v.31 no.2
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• pp.141-144
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• 1992

In this paper, some properties of p-injective ring is studied: The Jacobson radical of a pinjective ring which satisfies the ascending chain condition on essential left ideals is nilpotent. Also, the left singular ideal of a ring which satisfies the ascending chain condition on essential left ideals is nilpotent. Finally, we give an example which shows that a semiprime left p-injective ring such that every essential left ideal is two-sided is not necessarily to be strongly regular.egular.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.299-307
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• 2006

The following results ale extended from P-injective rings to AP-injective rings: (1) R is left self-injective regular if and only if R is a right (resp. left) AP-injective ring such that for every finitely generated left R-module M, $_R(M/Z(M))$ is projective, where Z(M) is the left singular submodule of $_{R}M$; (2) if R is a left nonsingular left AP-injective ring such that every maximal left ideal of R is either injective or a two-sided ideal of R, then R is either left self-injective regular or strongly regular. In addition, we answer a question of Roger Yue Chi Ming [13] in the positive. Let R be a ring whose every simple singular left R-module is Y J-injective. If R is a right MI-ring whose every essential right ideal is an essential left ideal, then R is a left and right self-injective regular, left and right V-ring of bounded index.

• Kyungpook Mathematical Journal
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• v.53 no.3
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• pp.397-406
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• 2013

A ring R is called a left (right) SF-ring if simple left (right) R-modules are flat. It is still unknown whether a left (right) SF-ring is von Neumann regular. In this paper, we give some conditions for a left (right) SF-ring to be (a) von Neumann regular; (b) strongly regular; (c) division ring. It is proved that: (1) a right SF-ring R is regular if maximal essential right (left) ideals of R are weakly left (right) ideals of R (this result gives an affirmative answer to the question raised by Zhang in 1994); (2) a left SF-ring R is strongly regular if every non-zero left (right) ideal of R contains a non-zero left (right) ideal of R which is a W-ideal; (3) if R is a left SF-ring such that $l(x)(r(x))$ is an essential left (right) ideal for every right (left) zero divisor x of R, then R is a division ring.

• East Asian mathematical journal
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• v.24 no.4
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• pp.377-387
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• 2008

We consider associative ring R (not necessarily commutative). In this paper the concepts: zero square ring of type-1/type-2, zero square ideal of type-1/type-2, zero square dimension of a ring R were introduced and obtained several important results. Finally, some relations between the zero square dimension of the direct sum of finite number of rings; and the sum of the zero square dimension of individual rings; were obtained. Necessary examples were provided.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.533-541
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• 2006

let E be a Finsler modules over $C^*$-algebras. A with norm-map $\rho$ and L(E) set of all A-linear bonded operators on E. We show that the canonical homomorphism ${\phi}:L(E){\rightarrow}L(E_I)$ sending each operator T to its restriction $T|E_I$ is injective if and only if I is an essential ideal in the underlying $C^*$-algebra A. We also show that $T{\in}L(E)$ is a bounded below if and only if ${\mid}{\mid}x{\mid}{\mid}={\mid}{\mid}{\rho}{\prime}(x){\mid}{\mid}$ is complete, where ${\rho}{\prime}(x)={\rho}(Tx)$ for all $x{\in}E$. Also, we give a necessary and sufficient condition for the equivalence of the norms generated by the norm map.

• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.37-43
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• 2008

The aim of the present paper is to introduce the concept "Finite dimension" in the theory of associative rings R with respect to two sided ideals. We obtain that if R has finite dimension on two sided ideals, then there exist uniform ideals $U_1,U_2,\ldots,U_n$ of R whose sum is direct and essential in R. The number n is independent of the choice of the uniform ideals $U_i$ and 'n' is called the dimension of R.

• East Asian mathematical journal
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• v.37 no.1
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• pp.33-40
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• 2021

We discuss the condition that if ab = 0 for elements a, b in a ring R then aIb = 0 for some essential ideal I of R. A ring with such condition is called IEIP. We prove that a ring R is IEIP if and only if Dn(R) is IEIP for every n ≥ 2, where Dn(R) is the ring of n by n upper triangular matrices over R whose diagonals are equal. We construct an IEIP ring that is not Abelian and show that a well-known Abelian ring is not IEIP, noting that rings with the insertion-of-factors-property are Abelian.