• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.867-871
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• 2009

A module M over a ring R is said to satisfy (P) if every generating set of M contains an independent generating set. The following results are proved; (1) Let $\tau$ = ($\mathbb{T}_\tau,\;\mathbb{F}_\tau$) be a hereditary torsion theory such that $\mathbb{T}_\tau$ $\neq$ Mod-R. Then every $\tau$-torsionfree R-module satisfies (P) if and only if S = R/$\tau$(R) is a division ring. (2) Let $\mathcal{K}$ be a hereditary pre-torsion class of modules. Then every module in $\mathcal{K}$ satisfies (P) if and only if either $\mathcal{K}$ = {0} or S = R/$Soc_\mathcal{K}$(R) is a division ring, where $Soc_\mathcal{K}$(R) = $\cap${I 4\leq$ $R_R$ : R/I$\in\mathcal{K}$}.

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

• Journal of Institute of Control, Robotics and Systems
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• v.21 no.11
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• pp.1027-1033
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• 2015

For the attitude control of spacecraft, two variable-speed control moment gyros are proposed as main actuators in the article. Since a variable-speed control moment gyro (VSCMG) makes two control torques (gyroscopic torque and reaction torque), two VSCMGs are sufficient for controlling 3-axes attitude. Additionally, there are no singular conditions for two non-parallel VSCMGs. Since gyroscopic torque is usually much greater than reaction torque, the control performances of approximately 3 axes may not be the same. However, several missions can be accomplished by controlling two axes. For such missions, a selective axes control method is proposed. The method selects two axes for a certain task and controls the attitude of the selected axes. For the remaining axis, angular speed is controlled for stabilization. A hardware-in-the-loop simulation has been used to test VSCMG modules and to verify the proposed method. Two VSCMGs can be alternative actuators for small satellites.