A ring R is called to be exchange if the right regular module RR has ÿnite exchange property. We ... more A ring R is called to be exchange if the right regular module RR has ÿnite exchange property. We continue in this paper the study of exchange rings by several authors. In particular, we investigate the von Neumann regularity of exchange rings. In addition, we also study whether the exchange property is inherited by some extensions of exchange rings.
Faith [5] raised the following questions: When does R being a right zip ring imply R[x] being rig... more Faith [5] raised the following questions: When does R being a right zip ring imply R[x] being right zip?; Characterize a ring R such that Mat n (R) is right zip; When does R being a right zip ring imply R[G] being right zip when G is a finite group? In this note, we continue the study of the extensions of noncommutative zip rings based on Faith's questions.
In the present note we study the Armendariz property on ideals of rings, introducing a new concep... more In the present note we study the Armendariz property on ideals of rings, introducing a new concept which unifies the Armendariz property and the insertion-of-factors-property (simply, IFP) for rings. In relation with this work, we investigate rings over which polynomial rings are IFP, called strongly IFP rings, which generalize both ideal-Armendariz rings and strongly reversible rings. The classes of minimal noncommutative ideal-Armendariz rings and strongly IFP rings, and the classes of minimal non-Abelian ideal-Armendariz rings and strongly IFP rings are completely determined, up to isomorphism. It is also shown that a local ring is Armendariz, symmetric, and strongly reversible (hence ideal-Armendariz) when the cardinality of the Jacobson radical is 4.
σ -Wedderburn radical σ -Levitzki radical Upper σ -nil radical We first introduce the σ -Wedderbu... more σ -Wedderburn radical σ -Levitzki radical Upper σ -nil radical We first introduce the σ -Wedderburn radical and the σ -Levitzki radical of a ring R, where σ is an automorphism of R. Using the properties of these radicals, we study the Wedderburn radical of the skew polynomial ring R[x; σ ] and the skew Laurent polynomial ring R[x, x -1 ; σ ], and next observe the Levitzki radical of R[x; σ ] and R[x, x -1 ; σ ]. Furthermore we characterize the upper nilradical of R[x; σ ] and R[x, x -1 ; σ ], via the upper σ -nil radical of R.
International Journal of Algebra and Computation, 2011
Jacobson introduced the concept of K-rings, continuing the investigation of Kaplansky and Herstei... more Jacobson introduced the concept of K-rings, continuing the investigation of Kaplansky and Herstein into the commutativity of rings. In this note we focus on the ring-theoretic properties of K-rings. We first construct basic examples of K-rings to be handled easily. It is shown that a semiprime K-ring of bounded index of nilpotency is a commutative domain. It is proved that if R is a prime K-ring then its classical quotient ring is a local ring with a nil Jacobson radical. We also show that if R is a π-regular K-ring then R/P is a field for every strongly prime ideal P of R. The basic structure of a condition, unifying K-rings and reversible rings, is studied with respect to zero-divisors in matrices and polynomials.
Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith... more Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith said a ring would be called strongly right bounded if every non-zero right ideal were bounded. In this paper we introduce a condition that is a generalisation of strongly bounded rings and insertion-of-factors-property (IFP) rings, calling a ring strongly right AB if every non-zero right annihilator is bounded. We first observe the structure of strongly right AB rings by analysing minimal non-commutative strongly right AB rings up to isomorphism. We study properties of strongly right AB rings, finding conditions for strongly right AB rings to be reduced or strongly right bounded. Relating to Ramamurthi's question (i.e. Are right and left SF rings von Neumann regular?), we show that a ring is strongly regular if and only if it is strongly right AB and right SF, from which we may generalise several known results. We also construct more examples of strongly right AB rings and counterex...
We observe a structure on the products of coefficients of nilpotent polynomials, introducing the ... more We observe a structure on the products of coefficients of nilpotent polynomials, introducing the concept of n-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classification of reduced rings, proving that a ring R is reduced if and only if the n by n upper triangular matrix ring over R is n-semi-Armendariz. It is shown that n-semi-Armendariz rings need not be (n+1)-semi-Armendariz and vice versa. We prove that a ring R is n-semi-Armendariz if and only if so is the polynomial ring over R. We next study interesting properties and useful examples of n-semi-Armendariz rings, constructing various kinds of counterexamples in the process.
Given a ring R, consider the condition: () every maximal right ideal of R contains a maximal idea... more Given a ring R, consider the condition: () every maximal right ideal of R contains a maximal ideal of R. We show that, for a ring R and 0 ≠ e = e ∈ R such that ele ⫋ eRe every proper ideal I of RR satisfies () if and only if eRe satisfies (). Hence with the help of some other results, () is a Morita invariant property. For a simple ring RR[x] satisfies () if and only if R[x] is not right primitive. By this result, if R is a division ring and R[x] satisfies (), then the Jacobson conjecture holds. We also show that for a finite centralizing extension S of a ring RR satisfies () if and only if S satisfies ().
A ring R is called to be exchange if the right regular module RR has ÿnite exchange property. We ... more A ring R is called to be exchange if the right regular module RR has ÿnite exchange property. We continue in this paper the study of exchange rings by several authors. In particular, we investigate the von Neumann regularity of exchange rings. In addition, we also study whether the exchange property is inherited by some extensions of exchange rings.
Faith [5] raised the following questions: When does R being a right zip ring imply R[x] being rig... more Faith [5] raised the following questions: When does R being a right zip ring imply R[x] being right zip?; Characterize a ring R such that Mat n (R) is right zip; When does R being a right zip ring imply R[G] being right zip when G is a finite group? In this note, we continue the study of the extensions of noncommutative zip rings based on Faith's questions.
In the present note we study the Armendariz property on ideals of rings, introducing a new concep... more In the present note we study the Armendariz property on ideals of rings, introducing a new concept which unifies the Armendariz property and the insertion-of-factors-property (simply, IFP) for rings. In relation with this work, we investigate rings over which polynomial rings are IFP, called strongly IFP rings, which generalize both ideal-Armendariz rings and strongly reversible rings. The classes of minimal noncommutative ideal-Armendariz rings and strongly IFP rings, and the classes of minimal non-Abelian ideal-Armendariz rings and strongly IFP rings are completely determined, up to isomorphism. It is also shown that a local ring is Armendariz, symmetric, and strongly reversible (hence ideal-Armendariz) when the cardinality of the Jacobson radical is 4.
σ -Wedderburn radical σ -Levitzki radical Upper σ -nil radical We first introduce the σ -Wedderbu... more σ -Wedderburn radical σ -Levitzki radical Upper σ -nil radical We first introduce the σ -Wedderburn radical and the σ -Levitzki radical of a ring R, where σ is an automorphism of R. Using the properties of these radicals, we study the Wedderburn radical of the skew polynomial ring R[x; σ ] and the skew Laurent polynomial ring R[x, x -1 ; σ ], and next observe the Levitzki radical of R[x; σ ] and R[x, x -1 ; σ ]. Furthermore we characterize the upper nilradical of R[x; σ ] and R[x, x -1 ; σ ], via the upper σ -nil radical of R.
International Journal of Algebra and Computation, 2011
Jacobson introduced the concept of K-rings, continuing the investigation of Kaplansky and Herstei... more Jacobson introduced the concept of K-rings, continuing the investigation of Kaplansky and Herstein into the commutativity of rings. In this note we focus on the ring-theoretic properties of K-rings. We first construct basic examples of K-rings to be handled easily. It is shown that a semiprime K-ring of bounded index of nilpotency is a commutative domain. It is proved that if R is a prime K-ring then its classical quotient ring is a local ring with a nil Jacobson radical. We also show that if R is a π-regular K-ring then R/P is a field for every strongly prime ideal P of R. The basic structure of a condition, unifying K-rings and reversible rings, is studied with respect to zero-divisors in matrices and polynomials.
Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith... more Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith said a ring would be called strongly right bounded if every non-zero right ideal were bounded. In this paper we introduce a condition that is a generalisation of strongly bounded rings and insertion-of-factors-property (IFP) rings, calling a ring strongly right AB if every non-zero right annihilator is bounded. We first observe the structure of strongly right AB rings by analysing minimal non-commutative strongly right AB rings up to isomorphism. We study properties of strongly right AB rings, finding conditions for strongly right AB rings to be reduced or strongly right bounded. Relating to Ramamurthi's question (i.e. Are right and left SF rings von Neumann regular?), we show that a ring is strongly regular if and only if it is strongly right AB and right SF, from which we may generalise several known results. We also construct more examples of strongly right AB rings and counterex...
We observe a structure on the products of coefficients of nilpotent polynomials, introducing the ... more We observe a structure on the products of coefficients of nilpotent polynomials, introducing the concept of n-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classification of reduced rings, proving that a ring R is reduced if and only if the n by n upper triangular matrix ring over R is n-semi-Armendariz. It is shown that n-semi-Armendariz rings need not be (n+1)-semi-Armendariz and vice versa. We prove that a ring R is n-semi-Armendariz if and only if so is the polynomial ring over R. We next study interesting properties and useful examples of n-semi-Armendariz rings, constructing various kinds of counterexamples in the process.
Given a ring R, consider the condition: () every maximal right ideal of R contains a maximal idea... more Given a ring R, consider the condition: () every maximal right ideal of R contains a maximal ideal of R. We show that, for a ring R and 0 ≠ e = e ∈ R such that ele ⫋ eRe every proper ideal I of RR satisfies () if and only if eRe satisfies (). Hence with the help of some other results, () is a Morita invariant property. For a simple ring RR[x] satisfies () if and only if R[x] is not right primitive. By this result, if R is a division ring and R[x] satisfies (), then the Jacobson conjecture holds. We also show that for a finite centralizing extension S of a ring RR satisfies () if and only if S satisfies ().
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