• Korean Journal of Mathematics
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• v.27 no.1
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• pp.141-150
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• 2019

In this article we investigate the structure of rings in which lower nilradicals coincide with upper nilradicals. Such rings shall be said to be quasi-2-primal. It is shown first that the $K{\ddot{o}}the^{\prime}s$ conjecture holds for quasi-2-primal rings. So the results in this article may provide interesting and useful information to the study of nilradicals in various situations. In the procedure we study the structure of quasi-2-primal rings, and observe various kinds of quasi-2-primal rings which do roles in ring theory.

• Bulletin of the Korean Mathematical Society
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• v.41 no.3
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• pp.507-519
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• 2004

For a ring endomorphism $\sigma$ of a ring R, J. Krempa called $\sigma$ a rigid endomorphism if a$\sigma$(a)＝0 implies a＝0 for a ${\in}$R. A ring R is called rigid if there exists a rigid endomorphism of R. In this paper, we extend the (J'-rigid property of a ring R to the upper nilradical $N_{r}$(R) of R. For an endomorphism R and the upper nilradical $N_{r}$(R) of a ring R, we introduce the condition (*): $N_{r}$(R) is a $\sigma$-ideal of R and a$\sigma$(a) ${\in}$ $N_{r}$(R) implies a ${\in}$ $N_{r}$(R) for a ${\in}$ R. We study characterizations of a ring R with an endomorphism $\sigma$ satisfying the condition (*), and we investigate their related properties. The connections between the upper nilradical of R and the upper nilradical of the skew power series ring R[[$\chi$;$\sigma$]] of R are also investigated.ated.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1309-1325
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• 2016

This note focuses on a ring property in which upper and lower nilradicals coincide, as a generalizations of symmetric rings. The concept of symmetric ideal and ring in the noncommutative ring theory was initially introduced by Lambek, as an extension of the usual commutative ideal theory. The investigation of symmetric rings provided many useful results to the study in the noncommutative ring theory. So the results obtained from this study may be applicable to observing the structure of zero divisors in various kinds of algebraic systems containing matrix rings and polynomial rings.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.511-515
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• 2012

We first show that the semiprimeness, primeness, and reducedness can go up to up-monoid rings. By these results we can compute the lower nilradicals of up-monoid rings, from which the well-known fact of Amitsur and McCoy for the polynomial rings can be extended to up-monoid rings.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.447-454
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• 2020

The studies of reversible and NI rings have done important roles in noncommutative ring theory. A ring R shall be called QRUR if ab = 0 for a, b ∈ R implies that ba is contained in the upper nilradical of R, which is a generalization of the NI ring property. In this article we investigate the structure of QRUR rings and examine the QRUR property of several kinds of ring extensions including matrix rings and polynomial rings. We also show that if there exists a weakly semicommutative ring but not QRUR, then Köthe's conjecture does not hold.

• Journal of the Korean Mathematical Society
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• v.51 no.6
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• pp.1251-1267
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• 2014

The semicommutative property of rings was introduced initially by Bell, and has done important roles in noncommutative ring theory. This concept was generalized to one of nil-semicommutative by Chen. We first study some basic properties of nil-semicommutative rings. We next investigate the structure of Ore extensions when upper nilradicals are ${\sigma}$-rigid ${\delta}$-ideals, examining the nil-semicommutative ring property of Ore extensions and skew power series rings, where ${\sigma}$ is a ring endomorphism and ${\delta}$ is a ${\sigma}$-derivation.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.27-39
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• 2021

We discuss the condition that every nonzero right annihilator of an element contains a nonzero ideal, as a generalization of the insertion-of-factors-property. A ring with such condition is called right AP. We prove that a ring R is right AP if and only if Dn(R) is right AP for every n ≥ 2, where Dn(R) is the ring of n by n upper triangular matrices over R whose diagonals are equal. Properties of right AP rings are investigated in relation to nilradicals, prime factor rings and minimal order.

• Korean Journal of Mathematics
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• v.24 no.4
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• pp.737-750
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• 2016

In this note we describe some classes of rings in relation to Abelian property of factorizations by nilradicals and Jacobson radical. The ring theoretical structures are investigated for various sorts of such factor rings which occur in the process.