nil-Armendariz Rings Research Articles

Nil-Armendariz Condition on Skew Generalized Power Series Rings

Let R be a ring, S a strictly ordered monoid, and $$\omega :S \to {\text{End}}(R)$$ a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Malcev–Neumann Laurent series rings. In the current work, we study the (S, ω)-nil Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials and power series to skew generalized power series. We resolve the structure of (S, ω)-nil Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S, ω)-nil Armendariz, unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special cases. For example, we show that left uniserial nilpotent semicommutative rings are nil Armendariz. Moreover, we study on the relationship between the zip and weak zip properties of a ring R and these of the ring R[[S, ω]].

Nilpotent Elements in Skew Polynomial Rings

Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We also prove that, if is nil--compatible and nil-Armendariz ring of power series type with nilpotent, then . We show that, if is a nil-Armendariz ring of power series type, with nilpotent and nil--compatible ring, then As a consequence, several known results are unified and extended to the more general setting. Also examples are provided to illustrate our results.

?-Armendariz Rings and Related Concepts

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.

Nilpotent elements and nil-Armendariz property of skew generalized power series rings

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid, and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. In this paper, we introduce and study the [Formula: see text]-nil-Armendariz condition on [Formula: see text], a generalization of the standard nil-Armendariz condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-nil-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-nil-Armendariz. The [Formula: see text]-nil-Armendariz condition is connected to the question of whether or not a skew generalized power series ring [Formula: see text] over a nil ring [Formula: see text] is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc. 7 (1956) 35–48]. As particular cases of our general results we obtain several new theorems on the nil-Armendariz condition. Our results extend and unify many existing results.

Nilradicals of the unique product monoid rings

Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring [Formula: see text] and a monoid [Formula: see text], we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring [Formula: see text]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.

π-Armendariz Rings and Related Concepts

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

On Monoid Rings Over Nil Armendariz Ring

Given a ring R and a monoid M, we study the concept of so called nil-Armendariz ring relative to a monoid, which is a common generalization of nil-Armendariz rings and Armendariz rings relative to a monoid. It is done by considering the nil-Armendariz condition on a monoid ring R[M] instead of the polynomial ring R[x]. We prove that several properties transfer between R and the monoid ring R[M], in case R is nil M-Armendariz ring. We resolve the structure of nil M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil M-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the special cases. In particular, we prove that every NI-ring is nil M-Armendariz, for any unique product monoid M. We also classify which of the standard nilpotence properties on polynomial rings pass to monoid rings. We provide various examples and classify how the nil M-Armendariz rings behaves under various ring extensions.

ARMENDARIZ PROPERTY OVER PRIME RADICALS

We observe from known results that the set of nilpotent elements in Armendariz 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[<TEX>$x$</TEX>], and that N(R[<TEX>$x$</TEX>]) = N(R)[<TEX>$x$</TEX>] when R is APR, where R[<TEX>$x$</TEX>] is the polynomial ring with an indeterminate <TEX>$x$</TEX> 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.

ON NILPOTENT POWER SERIES WITH NILPOTENT COEFFICIENTS

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, introducing the notion of nil-Armendariz rings. Hizem extended the nil-Armendariz property for polynomial rings onto power-series rings, say nil power-serieswise rings. In this paper, we introduce the notion of power-serieswise CN rings that is a generalization of nil power-serieswise Armendariz rings. Finally, we study the nil-Armendariz property for Ore extensions and skew power series rings.

Nilpotent elements and McCoy rings

We introduce the concept of nil-McCoy rings to study the structure of the set of nilpotent elements in McCoy rings. This notion extends the concepts of McCoy rings and nil-Armendariz rings. It is proved that every semicommutative ring is nil-McCoy. We shall give an example to show that nil-McCoy rings need not be semicommutative. Moreover, we show that nil-McCoy rings need not be right linearly McCoy. More examples of nil-McCoy rings are given by various extensions. On the other hand, the properties of α-McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x] are also investigated. For a monomorphism α of a ring R, it is shown that if R is weak α-rigid and α-reversible then R is α-McCoy.

NIL-ARMENDARIZ RINGS AND UPPER NILRADICALS

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.

NILPOTENT ELEMENTS AND NIL-ARMENDARIZ PROPERTY OF MONOID RINGS

Antoine [Nilpotent elements and Armendariz rings, J. Algebra 319(8) (2008) 3128–3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. For a monoid M, we introduce nil-Armendariz rings relative to M, which is a generalization of nil-Armendariz rings and we investigate their properties. This condition is strongly connected to the question of whether or not a monoid ring R[M] over a nil ring R is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc.7 (1956) 35–48]. This is true for any 2-primal ring R and u.p.-monoid M. If the set of nilpotent elements of a ring R forms an ideal, then R is nil-Armendariz relative to any u.p.-monoid M. Also, for any monoid M with an element of infinite order, M-Armendariz rings are nil M-Armendariz. When R is a 2-primal ring, then R[x] and R[x, x-1] are nil-Armendariz relative to any u.p.-monoid M, and we have nil (R[M]) = nil (R)[M].

Nil-Armendariz Rings Relative to a Monoid

For a monoid M, we introduce nil-Armendariz rings relative to M, which are a generalization of nil-Armendariz and M-Armendariz rings, and investigate their properties. First we show that semicommutative rings are nil-Armendariz relative to every unique product monoid M. Also it is shown that for a strictly totally ordered monoid M and an ideal I of R, if I is a semicommutative subrng of R and R/I nil-Armendariz relative to M, then R is nil-Armendariz relative to M. Then we show that if R is a semicommutative ring and nil-Armendariz relative to M, then R is nil-Armendariz relative to M × N, where N is a unique product monoid. As corollaries we obtain some results of [2] and [10].

ON NIL SKEW ARMENDARIZ RINGS

Antoine [Nilpotent elements and Armendariz rings, J. Algebra319 (2008) 3128–3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. When the set of nilpotent elements of a ring R with an α-condition, namely α-compatibility, forms an ideal, we observe that R satisfies a nil Armendariz-type property, in the context of Ore extension R[x;α, δ]. For a 2-primal ring R with a derivation δ, R[x] is nil [Formula: see text]-skew Armendariz, and for a 2-primal ring R, R is nil α-skew Armendariz if and only if R[x] is nil [Formula: see text]-skew Armendariz, where α is an endomorphism of R with αk = id R, for some positive integer k. Moreover, we prove that a ring R is nil (α, δ)-skew Armendariz if and only if the n-by-n skew triangular matrix ring Tn(R, σ) is nil [Formula: see text]-skew Armendariz, for each endomorphism σ, with σ(1) = 1. A rich source of rings R, for which R[x] is nil [Formula: see text]-skew Armendariz, is provided.

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

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.