On conditions provided by Nilradicals

김홍기,김남균,정문섭,이양,류성주,여동은 대한수학회 2009 대한수학회지 Vol.46 No.5

A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for a, b∈ 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. A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for a, b∈ 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.

ON JACOBSON AND NIL RADICALS RELATED TO POLYNOMIAL RINGS

Kwak, Tai Keun,Lee, Yang,Ozcan, A. Cigdem Korean Mathematical Society 2016 대한수학회지 Vol.53 No.2

This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that J(R[x]) = J(R)[x] if and only if J(R) is nil when a given ring R is Armendariz, where J(A) means the Jacobson radical of a ring A. A ring will be called feckly Armendariz if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.

On Jacobson and nil radicals related to polynomial rings

곽태근,이양,A. Cigdem Ozcan 대한수학회 2016 대한수학회지 Vol.53 No.2

This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that $J(R[x])=J(R)[x]$ if and only if $J(R)$ is nil when a given ring $R$ is Armendariz, where $J(A)$ means the Jacobson radical of a ring $A$. A ring will be called {\it feckly Armendariz} if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.

QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS

Baser, Muhittin,Kwa, Tai Keun Korean Mathematical Society 2011 대한수학회논문집 Vol.26 No.4

The concept of the quasi-Armendariz property of rings properly contains Armendariz rings and semiprime rings. In this paper, we extend the quasi-Armendariz property for a polynomial ring to the skew polynomial ring, hence we call such ring a ${\sigma}$-quasi-Armendariz ring for a ring endomorphism ${\sigma}$, and investigate its structures, several extensions and related properties. In particular, we study the semiprimeness and the quasi-Armendariz property between a ring R and the skew polynomial ring R[x;${\sigma}$$] of R, and so these provide us with an opportunity to study quasi-Armendariz rings and semiprime rings in a general setting, and several known results follow as consequences of our results.

Power Series Rings Satisfying a Zero Divisor Property

Kim, Nam,Lee, Ki,Lee, Yang Taylor Francis 2006 Communications in Algebra Vol.34 No.6

<P>In this note we continue to study zero divisors in power series rings and polynomial rings over general noncommutative rings. We first construct Armendariz rings which are not power-serieswise Armendariz, and find various properties of (power-serieswise) Armendariz rings. We show that for a semiprime power-serieswise Armendariz (so reduced) ring R with a.c.c. on annihilator ideals, R [[ x ]] (the power series ring with an indeterminate x over R ) has finitely many minimal prime ideals, say B 1 ,…, B m , such that B 1 … B m = 0 and B i = A i [[ x ]] for some minimal prime ideal A i of R for all i , where A 1 ,…, A m are all minimal prime ideals of R . We also prove that the power-serieswise Armendarizness is preserved by the polynomial ring extension as the Armendarizness, and construct various types of (power-serieswise) Armendariz rings.</P>

ANNIHILATING CONTENT IN POLYNOMIAL AND POWER SERIES RINGS

Abuosba, Emad,Ghanem, Manal Korean Mathematical Society 2019 대한수학회지 Vol.56 No.5

Let R be a commutative ring with unity. If f(x) is a zero-divisor polynomial such that $f(x)=c_f f_1(x)$ with $c_f{\in}R$ and $f_1(x)$ is not zero-divisor, then $c_f$ is called an annihilating content for f(x). In this case $Ann(f)=Ann(c_f )$. We defined EM-rings to be rings with every zero-divisor polynomial having annihilating content. We showed that the class of EM-rings includes integral domains, principal ideal rings, and PP-rings, while it is included in Armendariz rings, and rings having a.c. condition. Some properties of EM-rings are studied and the zero-divisor graphs ${\Gamma}(R)$ and ${\Gamma}(R[x])$ are related if R was an EM-ring. Some properties of annihilating contents for polynomials are extended to formal power series rings.

Annihilating content in polynomial and power series rings

Emad Abu Osba,Manal Ghanem 대한수학회 2019 대한수학회지 Vol.56 No.5

Let $R$ be a commutative ring with unity. If $f(x)$ is a zero-divisor polynomial such that $f(x)=c_{f}f_{1}(x)$ with $c_{f}\in R$ and $f_{1}(x)$ is not zero-divisor, then $c_{f}$ is called an annihilating content for $ f(x) $. In this case $Ann(f)=Ann(c_{f})$. We defined EM-rings to be rings with every zero-divisor polynomial having annihilating content. We showed that the class of EM-rings includes integral domains, principal ideal rings, and PP-rings, while it is included in Armendariz rings, and rings having a.c. condition. Some properties of EM-rings are studied and the zero-divisor graphs $\Gamma (R)$ and $\Gamma (R[x])$ are related if $R$ was an EM-ring. Some properties of annihilating contents for polynomials are extended to formal power series rings.

Rings with Property (A) and their extensions

Hong, C.Y.,Kim, N.K.,Lee, Y.,Ryu, S.J. Academic Press 2007 Journal of algebra Vol.315 No.2

A commutative ring R has Property (A) if every finitely generated ideal of R consisting entirely of zero-divisors has a nonzero annihilator. We continue in this paper the study of rings with Property (A). We extend Property (A) to noncommutative rings, and study such rings. Moreover, we study several extensions of rings with Property (A) including matrix rings, polynomial rings, power series rings and classical quotient rings. Finally, we characterize when the space of minimal prime ideals of rings with Property (A) is compact.

RINGS OVER WHICH POLYNOMIAL RINGS ARE ARMENDARIZ AND REVERSIBLE

Ahn, Jung Ho,Choi, Min Jeong,Choi, Si Ra,Jeong, Won Seok,Kim, Jung Soo,Lee, Jeong Yeol,Lee, Soon Ji,Lee, Young Sun,Noh, Dong Hyun,Noh, Yu Seung,Park, Gyeong Hyeon,Lee, Chang Ik,Lee, Yang The Kangwon-Kyungki Mathematical Society 2012 한국수학논문집 Vol.20 No.3

A ring R is called reversibly Armendariz if $b_ja_i=0$ for all $i$, $j$ whenever $f(x)g(x)=0$ for two polynomials $f(x)=\sum_{i=0}^{m}a_ix^i,\;g(x)=\sum_{j=0}^{n}b_jx^j$ over R. It is proved that a ring R is reversibly Armendariz if and only if its polynomial ring is reversibly Armendariz if and only if its Laurent polynomial ring is reversibly Armendariz. Relations between reversibly Armendariz rings and related ring properties are examined in this note, observing the structures of many examples concerned. Various kinds of reversibly Armendariz rings are provided in the process. Especially it is shown to be possible to construct reversibly Armendariz rings from given any Armendariz rings.

Armendariz property over prime radicals

한준철,김홍기,이양 대한수학회 2013 대한수학회지 Vol.50 No.5

We observe from known results that the set of nilpotent elements inArmendariz rings has an important role. The upper nilradical coincides with the prime radical in Armendariz rings. So it can be shown that the factor ring of an Armendariz ring over its prime radical is also Armendariz, with the help of Antoine's results for nil-Armendariz rings. We study the structure of rings with such property in Armendariz rings and introduce APR as a generalization. It is shown that APR is placed between Armendariz and nil-Armendariz. It is shown that an APR ring which is not Armendariz, can always be constructed from any Armendariz ring. It is also proved that a ring R is APR if and only if so is R[x], and that N(R[x])=N(R)[x] when R is APR, where R[x] is the polynomial ring with an indeterminate x over R and N(-) denotes the set of all nilpotent elements. Several kinds of APR rings are found or constructed in the precess related to ordinary ring constructions.