• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.87-94
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• 2007

We in this note introduce the concept of g-IFP rings which is a generalization of IFP rings. We show that from any IFP ring there can be constructed a right g-IFP ring but not IFP. We also study the basic properties of right g-IFP rings, constructing suitable examples to the situations raised naturally in the process.

• Korean Journal of Mathematics
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• v.23 no.1
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• pp.37-46
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• 2015

We in this note introduce the concept of semi-IFP rings which is a generalization of IFP rings. We study the basic structure of semi-IFP rings, and construct suitable examples to the situations raised naturally in the process. We also show that the semi-IFP does not go up to polynomial rings.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1161-1178
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• 2015

In this paper, we investigate the insertion-of-factors-property (simply, IFP) on skew polynomial rings, introducing the concept of strongly ${\sigma}-IFP$ for a ring endomorphism ${\sigma}$. A ring R is said to have strongly ${\sigma}-IFP$ if the skew polynomial ring R[x;${\sigma}$] has IFP. We examine some characterizations and extensions of strongly ${\sigma}-IFP$ rings in relation with several ring theoretic properties which have important roles in ring theory. We also extend many of related basic results to the wider classes, and so several known results follow as consequences of our results.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.727-740
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• 2008

A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for $a,b{\in}R$. Huh et al. showed that the IFP condition need not be preserved by polynomial ring extensions. But it is shown that ${\sum}^n_{i=0}$ $E_{ai}E$ is a nonzero nilpotent ideal of E whenever R is an IFP ring and $0{\neq}f{\in}F$ is nilpotent, where E is a polynomial ring over R, F is a polynomial ring over E, and $a_i^{'s}$ are the coefficients of f. we shall use the term near IFP to denote such a ring as having place near at the IFPness. In the present note the structures of IFP rings and near-IFP rings are observed, extending the classes of them. IFP rings are NI (i.e., nilpotent elements form an ideal). It is shown that the near-IFPness and the NIness are distinct each other, and the relations among them and related conditions are examined.

• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.1027-1040
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• 2009

A ring R is called IFP, due to Bell, if ab = 0 implies aRb = 0 for a, b $\in$ R. Huh et al. showed that the IFP condition is not preserved by polynomial ring extensions. In this note we concentrate on a generalized condition of the IFPness that can be lifted up to polynomial rings, introducing the concept of quasi-IFP rings. The structure of quasi-IFP rings will be studied, characterizing quasi-IFP rings via minimal strongly prime ideals. The connections between quasi-IFP rings and related concepts are also observed in various situations, constructing necessary examples in the process. The structure of minimal noncommutative (quasi-)IFP rings is also observed.

• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.381-394
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• 2012

We generalize the insertion-of-factors-property by setting nilpotent products of elements. In the process we introduce the concept of a nil-IFP ring that is also a generalization of an NI ring. It is shown that if K$\ddot{o}$the's conjecture holds, then every nil-IFP ring is NI. The class of minimal noncommutative nil-IFP rings is completely determined, up to isomorphism, where the minimal means having smallest cardinality.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1389-1396
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• 2018

We observe the structure of a kind of unit-IFP ring that is constructed by Antoine, in relation with units and nilpotent elements. This article concerns the same argument in a more general situation, and study the structure of one-sided zero divisors in such rings. We also provide another kind of unit-IFP ring.

• Kyungpook Mathematical Journal
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• v.62 no.2
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• pp.213-227
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• 2022

We discuss the relation between units and nilpotents of a ring, concentrating on the transitivity of units on nilpotents under regular group actions. We first prove that for a ring R, if U(R) is right transitive on N(R), then Köthe's conjecture holds for R, where U(R) and N(R) are the group of all units and the set of all nilpotents in R, respectively. A ring is called right UN-transitive if it satisfies this transitivity, as a generalization, a ring is called unilpotent-IFP if aU(R) ⊆ N(R) for all a ∈ N(R). We study the structures of right UN-transitive and unilpotent-IFP rings in relation to radicals, NI rings, unit-IFP rings, matrix rings and polynomial rings.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1257-1268
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• 2018

In this article we first investigate a sort of unit-IFP ring by which Antoine provides very useful information to ring theory in relation with the structure of coefficients of zero-dividing polynomials. Here we are concerned with the whole shape of units and nilpotent elements in such rings. Next we study the properties of unit-IFP rings through group actions of units on nonzero nilpotent elements. We prove that if R is a unit-IFP ring such that there are finite number of orbits under the left (resp., right) action of units on nonzero nilpotent elements, then R satisfies the descending chain condition for nil left (resp., right) ideals of R and the upper nilradical of R is nilpotent.

• Kyungpook Mathematical Journal
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• v.60 no.2
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• pp.307-318
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• 2020

This article concerns the class of rings which satisfy the property of inserting regular elements at zero products, and rings with such property are called regular-IFP. We study the structure of regular-IFP rings in relation to various ring properties that play roles in noncommutative ring theory. We investigate conditions under which the regular-IFPness pass to polynomial rings, and equivalent conditions to the regular-IFPness.