ON Π-NEAR-ARMENDARIZ RINGS

In this note we introduce a concept, so-called π-Near-Armendariz ring, that is a generalization of both Armendariz rings and 2-primal rings. We first observe the basic properties of π-Near-Armendariz rings, constructing typical examples. We next extend the class of π-Near-Armendariz rings, through various ring extensions.

Generalizations of reversible and Armendariz rings

International Journal of Algebra and Computation ◽

10.1142/s0218196716500387 ◽

2016 ◽

Vol 26 (05) ◽

pp. 911-933

Author(s):

Juncheol Han ◽

Tai Keun Kwak ◽

Chang Ik Lee ◽

Yang Lee ◽

Yeonsook Seo

Keyword(s):

Basic Properties ◽

Armendariz Rings ◽

This paper concerns several ring theoretic properties related to matrices and polynomials. The basic properties of [Formula: see text]-reversible and power-Armendariz are studied. We provide a method by which one can always construct a power-Armendariz ring but neither symmetric nor Armendariz from given any symmetric ring. We investigate next various interesting relations among ring theoretic properties containing [Formula: see text]-reversibility and power-Armendariz condition.

ANNIHILATOR CONDITIONS IN MATRIX AND SKEW POLYNOMIAL RINGS

10.1142/s021949881250079x ◽

2012 ◽

Vol 11 (04) ◽

pp. 1250079 ◽

Cited By ~ 7

Author(s):

A. ALHEVAZ ◽

A. MOUSSAVI

Keyword(s):

Polynomial Rings ◽

Trivial Extension ◽

Skew Polynomial Ring ◽

Matrix Rings ◽

Skew Polynomial Rings ◽

Armendariz Rings ◽

Ore Extensions ◽

Necessary And Sufficient ◽

Skew Polynomial ◽

Let R be a ring with an endomorphism α and α-derivation δ. By [A. R. Nasr-Isfahani and A. Moussavi, Ore extensions of skew Armendariz rings, Comm. Algebra 36(2) (2008) 508–522], a ring R is called a skew Armendariz ring, if for polynomials f(x) = a0 + a1 x + ⋯ + anxn, g(x) = b0+b1x + ⋯ + bmxm in R[x; α, δ], f(x)g(x) = 0 implies a0bj = 0 for each 0 ≤ j ≤ m. In this paper, radicals of the skew polynomial ring R[x; α, δ], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; α, δ], in case R is an α-compatible skew Armendariz ring. We also identify some "relatively maximal" skew Armendariz subrings of matrix rings, and obtain a necessary and sufficient condition for a trivial extension to be skew Armendariz. Consequently, new families of non-reduced skew Armendariz rings are presented and several known results related to Armendariz rings and skew polynomial rings will be extended and unified.

On Radicals of Skew Inverse Laurent Series Rings

Algebra Colloquium ◽

10.1142/s1005386716000353 ◽

2016 ◽

Vol 23 (02) ◽

pp. 335-346

Author(s):

A. Moussavi

Keyword(s):

Polynomial Ring ◽

Jacobson Radical ◽

Laurent Series ◽

Ring Extensions ◽

Series Type ◽

Let R be a ring and α an automorphism of R. Amitsur proved that the Jacobson radical J(R[x]) of the polynomial ring R[x] is the polynomial ring over the nil ideal J(R[x]) ∩ R. Following Amitsur, it is shown that when R is an Armendariz ring of skew inverse Laurent series type and S is any one of the ring extensions R[x;α], R[x,x-1;α], R[[x-1;α]] and R((x-1;α)), then ℜ𝔞𝔡(S) = ℜ𝔞𝔡(R)S = Nil (S), ℜ𝔞𝔡(S) ∩ R = Nil (R), where ℜ𝔞𝔡 is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals.

Commutative rings and modules that are Nil-coherent or special Nil-coherent

10.1142/s0219498817501870 ◽

2017 ◽

Vol 16 (10) ◽

pp. 1750187 ◽

Cited By ~ 2

Author(s):

Karima Alaoui Ismaili ◽

David E. Dobbs ◽

Najib Mahdou

Keyword(s):

Commutative Rings ◽

Finitely Generated ◽

Basic Properties ◽

Unital Ring ◽

Reduced Ring ◽

Coherent Ring ◽

Ring Extensions ◽

Coherent Rings ◽

Finitely Presented

Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

International Journal of Algebra and Computation ◽

10.1142/s0218196711006431 ◽

2011 ◽

Vol 21 (05) ◽

pp. 745-762 ◽

Cited By ~ 15

Author(s):

TAI KEUN KWAK ◽

YANG LEE

Keyword(s):

Polynomial Ring ◽

Regular Ring ◽

Von Neumann Regular Ring ◽

Von Neumann ◽

Nilpotent Elements ◽

Von Neumann Regular ◽

Armendariz Rings ◽

Ring Extensions ◽

Nil Ring

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

On α-nilpotent elements and α-Armendariz rings

10.1142/s0219498815500644 ◽

2015 ◽

Vol 14 (05) ◽

pp. 1550064

Author(s):

Hong Kee Kim ◽

Nam Kyun Kim ◽

Tai Keun Kwak ◽

Yang Lee ◽

Hidetoshi Marubayashi

Keyword(s):

Polynomial Ring ◽

Polynomial Rings ◽

Skew Polynomial Ring ◽

Nilpotent Elements ◽

Skew Polynomial Rings ◽

Armendariz Rings ◽

Skew Polynomial ◽

Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced the concept of nil-Armendariz property as a generalization. Hong et al. studied Armendariz property on skew polynomial rings and introduced the notion of an α-Armendariz ring, where α is a ring monomorphism. In this paper, we investigate the structure of the set of α-nilpotent elements in α-Armendariz rings and introduce an α-nil-Armendariz ring. We examine the set of [Formula: see text]-nilpotent elements in a skew polynomial ring R[x;α], where [Formula: see text] is the monomorphism induced by the monomorphism α of an α-Armendariz ring R. We prove that every polynomial with α-nilpotent coefficients in a ring R is [Formula: see text]-nilpotent when R is of bounded index of α-nilpotency, and moreover, R is shown to be α-nil-Armendariz in this situation. We also characterize the structure of the set of α-nilpotent elements in α-nil-Armendariz rings, and investigate the relations between α-(nil-)Armendariz property and other standard ring theoretic properties.

Further results on central Armendariz rings

10.1142/s0219498817501948 ◽

2017 ◽

Vol 16 (10) ◽

pp. 1750194 ◽

Cited By ~ 2

Author(s):

Weixing Chen

Keyword(s):

Prime Radical ◽

Armendariz Rings ◽

A ring [Formula: see text] is central Armendariz if [Formula: see text] and [Formula: see text] [Formula: see text] over [Formula: see text] satisfy [Formula: see text] then all [Formula: see text] are central. It is proved that if [Formula: see text] is a central Armendariz ring, then [Formula: see text] implies that all [Formula: see text] are in its prime radical.

Properties of Armendariz rings and weak Armendariz rings

Publications de l Institut Mathematique ◽

10.2298/pim0999131j ◽

2009 ◽

Vol 85 (99) ◽

pp. 131-137 ◽

Cited By ~ 1

Author(s):

Dusan Jokanovic

Keyword(s):

Power Series ◽

Direct Product ◽

Nilpotent Ideal ◽

Factor Ring ◽

Ring Isomorphism ◽

Armendariz Rings ◽

We consider some properties of Armendariz and rigid rings. We prove that the direct product of rigid (weak rigid), weak Armendariz rings is a rigid (weak rigid), weak Armendariz ring. On the assumption that the factor ring R/I is weak Armendariz, where I is nilpotent ideal, we prove that R is a weak Armendariz ring. We also prove that every ring isomorphism preserves weak skew Armendariz structure. Armendariz rings of Laurent power series are also considered.

NIL-ARMENDARIZ RINGS AND UPPER NILRADICALS

International Journal of Algebra and Computation ◽

10.1142/s0218196712500592 ◽

2012 ◽

Vol 22 (06) ◽

pp. 1250059 ◽

Cited By ~ 6

Author(s):

DA WOON JUNG ◽

NAM KYUN KIM ◽

YANG LEE ◽

SUNG PIL YANG

Keyword(s):

Regular Ring ◽

Direct Method ◽

Von Neumann Regular Ring ◽

Von Neumann ◽

Von Neumann Regular ◽

Armendariz Rings ◽

We continue the study of nil-Armendariz rings, initiated by Antoine, and Armendariz rings. We first examine a kind of ring coproduct constructed by Antoine for which the Armendariz, nil-Armendariz, and weak Armendariz properties are equivalent. Such a ring has an important role in the study of Armendariz ring property and near-related ring properties. We next prove an Antoine's result in relation with the ring coproduct by means of a simpler direct method. In the proof we can observe the concrete shapes of coefficients of zero-dividing polynomials. We next observe the structure of nil-Armendariz rings via the upper nilradicals. It is also shown that a ring R is Armendariz if and only if R is nil-Armendariz if and only if R is weak Armendariz, when R is a von Neumann regular ring.

?-Armendariz Rings and Related Concepts

Baghdad Science Journal ◽

10.21123/bsj.13.4.853-861 ◽

2016 ◽

Vol 13 (4) ◽

pp. 853-861

Author(s):

Baghdad Science Journal

Keyword(s):

Armendariz Rings ◽

In this paper we investigated some new properties of ?-Armendariz rings and studied the relationships between ?-Armendariz rings and central Armendariz rings, nil-Armendariz rings, semicommutative rings, skew Armendariz rings, ?-compatible rings and others. We proved that if R is a central Armendariz, then R is ?-Armendariz ring. Also we explained how skew Armendariz rings can be ?-Armendariz, for that we proved that if R is a skew Armendariz?-compatible ring, then R is ?-Armendariz. Examples are given to illustrate the relations between concepts.