• Communications of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.711-719
• /
• 2018

In this paper we define and study a new kind of dimension called, semisimple dimension, that measures how far a module is from being semisimple. Like other kinds of dimensions, this is an ordinal valued invariant. We give some interesting and useful properties of rings or modules which have semisimple dimension. It is shown that a noetherian module with semisimple dimension is an artinian module. A domain with semisimple dimension is a division ring. Also, for a semiprime right non-singular ring R, if its maximal right quotient ring has semisimple dimension as a right R-module, then R is a semisimple artinian ring. We also characterize rings whose modules have semisimple dimension. In fact, it is shown that all right R-modules have semisimple dimension if and only if the free right R-module ${\oplus}^{\infty}_{i=1}$ R has semisimple dimension, if and only if R is a semisimple artinian ring.

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.5
• /
• pp.1317-1325
• /
• 2022

In this note, we prove that an Artinian local ring is G-semisimple (resp., SG-semisimple, 2-SG-semisimple) if and only if its maximal ideal is G-projective (resp., SG-projective, 2-SG-projective). As a corollary, we obtain the global statement of the above. We also give some examples of local G-semisimple rings whose maximal ideals are n-generated for some positive integer n.

• Kyungpook Mathematical Journal
• /
• v.62 no.4
• /
• pp.641-655
• /
• 2022

A right R-module N is called pseudo semisimple-M-injective if for any monomorphism from every semisimple submodule of M to N, can be extended to a homomorphism from M to N. In this paper, we study some properties of pseudo semisimple-injective modules. Moreover, some results of pseudo semisimple-injective modules over formal triangular matrix rings are obtained.

• Communications of the Korean Mathematical Society
• /
• v.10 no.4
• /
• pp.839-847
• /
• 1995

For the given torsion theory $\tau$, we study some equivalent conditions when the localized ring $R_\tau$ be semisimple artinian (Theorem 4). Using this, if $R_\tau$ is semisimple artinian ring, we study when does the given ring R become left V-ring?

• Journal of applied mathematics & informatics
• /
• v.41 no.4
• /
• pp.831-838
• /
• 2023

In this paper, we define the concept of a semisimple injective comodule. We show that it is a generalization of some known comodules, and we give a useful characterization of this notion. Finally, we obtain a new characterization of simple semiartinian coring using semisimple injective comodules.

• Kyungpook Mathematical Journal
• /
• v.50 no.3
• /
• pp.411-417
• /
• 2010

Some characterizations of regular $\Gamma$-rings are described by means of fuzzy ideals. The concepts of fuzzy interior ideals in $\Gamma$-rings and semisimple $\Gamma$-rings are introduced. Some characterizations of semisimple $\Gamma$-rings are investigated by means of fuzzy interior ideals.

• Journal of applied mathematics & informatics
• /
• v.9 no.2
• /
• pp.859-867
• /
• 2002

In this paper, we get a necessary and sufficient condition for an inner automorphism between compact connected semisimple Lie groups to be an atone transformation, and obtain atone transformations of (SU(n),g) with some left invariant metric g.

• Bulletin of the Korean Mathematical Society
• /
• v.35 no.3
• /
• pp.583-590
• /
• 1998

The purpose of this paper is to prove the following results: Let A be a noncommutative semisimple Banach algebra. (1) Suppose that a linear derivation D : A $\to A$ is such that [D(x),x]x=0 holds for all $x \in A$. Then we have D=0. (2) Suppose that a linear derivation $D:A\to A$ is such that $D(x)x^2 + x^2D(x)=0$ holds for all $x \in A$. Then we have C=0.

• Communications of the Korean Mathematical Society
• /
• v.11 no.3
• /
• pp.595-600
• /
• 1996

We give a characterization of a deductive system. We introduce the concept of maximal deductive systems and show that every bounded Hilbert algebra with at least two elements contains at least one maximal deductive system. Moreover, we introduce the notion of radical and semisimple in a Hilbert algebra and prove that if H is a bounded Hilbert algebra in which every element is an involution, then H is semisimple.

• Communications of the Korean Mathematical Society
• /
• v.32 no.3
• /
• pp.745-763
• /
• 2017

The Borel-Serre partial compactification gives cofinite models for the proper classifying space for arithmetic lattices. Non-arithmetic lattices arise only in semisimple Lie groups of ${\mathbb{R}}$-rank one. The author generalizes the Borel-Serre partial compactification to construct cofinite models for the proper classifying space for lattices in semisimple Lie groups of ${\mathbb{R}}$-rank one by using the reduction theory of Garland and Raghunathan.