McCoy Rings Research Articles

MCCOY CONDITION ON IDEALS OF COEFFICIENTS

We continue the study of McCoy condition to analyze zero-dividing polynomials for the constant annihilators in the ideals generated by the coefficients. In the process we introduce the concept of ideal-<TEX>${\pi}$</TEX>-McCoy rings, extending known results related to McCoy condition. It is shown that the class of ideal-<TEX>${\pi}$</TEX>-McCoy rings contains both strongly McCoy rings whose non-regular polynomials are nilpotent and 2-primal rings. We also investigate relations between the ideal-<TEX>${\pi}$</TEX>-McCoy property and other standard ring theoretic properties. Moreover we extend the class of ideal-<TEX>${\pi}$</TEX>-McCoy rings by examining various sorts of ordinary ring extensions.

On Zero-Divisors in Skew Inverse Laurent Series Over NonCommutative Rings

In the present note, we continue to study zero-divisor properties of general skew inverse Laurent series rings R((x −1; σ, δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ. Extending the results of a large number of papers on the McCoy property for ordinary and skew polynomial rings, we study a version of the McCoy property for general skew inverse Laurent series extensions. We obtain some necessary or sufficient conditions for a ring to be (σ, δ)-SILS McCoy and prove that this property is preserved under a number of ring extensions. In particular, it passes to certain subrings of the (skew-)upper triangular matrix rings. In relation with this work and as an application of (σ, δ)-SILS McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew inverse Laurent series ring R((x −1; σ, δ)) and the graph-theoretical properties of its zero-divisor graph .

McCOY PROPERTY OF SKEW LAURENT POLYNOMIALS AND POWER SERIES RINGS

One of the important properties of commutative rings, proved by McCoy [Remarks on divisors of zero, Amer. Math. Monthly49(5) (1942) 286–295], is that if two nonzero polynomials annihilate each other over a commutative ring then each polynomial has a nonzero annihilator in the base ring. Nielsen [Semi-commutativity and the McCoy condition, J. Algebra298(1) (2006) 134–141] generalizes this property to non-commutative rings. Let M be a monoid and σ be an automorphism of a ring R. For the continuation of McCoy property of non-commutative rings, in this paper, we extend the McCoy's theorem to skew Laurent power series ring R[[x, x-1; σ]] and skew monoid ring R * M over general non-commutative rings. Constructing various examples, we classify how these skew versions of McCoy property behaves under various ring extensions. Moreover, we investigate relations between these properties and other standard ring-theoretic properties such as zip rings and rings with Property (A). As a consequence we extend and unify several known results related to McCoy rings.

Extensions of linearly McCoy rings

A ring R is called linearly McCoy if whenever linear polynomials <TEX>$f(x)$</TEX>, <TEX>$g(x){\in}R[x</TEX><TEX>]</TEX><TEX>{\backslash}\{0\}$</TEX> satisfy <TEX>$f(x)g(x)=0$</TEX>, there exist nonzero elements <TEX>$r,s{\in}R$</TEX> such that <TEX>$f(x)r=sg(x)=0$</TEX>. In this paper, extension properties of linearly McCoy rings are investigated. We prove that the polynomial ring over a linearly McCoy ring need not be linearly McCoy. It is shown that if there exists the classical right quotient ring Q of a ring R, then R is right linearly McCoy if and only if so is Q. Other basic extensions are also considered.

ON A RING PROPERTY UNIFYING REVERSIBLE AND RIGHT DUO RINGS

The concepts of reversible, right duo, and Armendariz rings are known to play important roles in ring theory and they are indepen- dent of one another. In this note we focus on a concept that can unify them, calling it a right Armendarizlike ring in the process. We first find a simple way to construct a right Armendarizlike ring but not Armendariz (reversible, or right duo). We show the difference between right Armen- darizlike rings and strongly right McCoy rings by examining the structure of right annihilators. For a regular ring R, it is proved that R is right Armendarizlike if and only if R is strongly right McCoy if and only if R is Abelian (entailing that right Armendarizlike, Armendariz, reversible, right duo, and IFP properties are equivalent for regular rings). It is shown that a ring R is right Armendarizlike, if and only if so is the polynomial ring over R, if and only if so is the classical right quotient ring (if any). In the process necessary (counter)examples are found or constructed. 1. Right Armendarizlike rings

The McCoy Condition on Ore Extensions

Nielsen [29] proved that all reversible rings are McCoy and gave an example of a semicommutative ring that is not right McCoy. When R is a reversible ring with an (α, δ)-condition, namely (α, δ)-compatibility, we observe that R satisfies a McCoy-type property, in the context of Ore extension R[x; α, δ], and provide rich classes of reversible (semicommutative) (α, δ)-compatible rings. It is also shown that semicommutative α-compatible rings are linearly α-skew McCoy and that linearly α-skew McCoy rings are Dedekind finite. Moreover, several extensions of skew McCoy rings and the zip property of these rings are studied.

α-Skewπ-McCoy Rings

As a generalization ofα-skew McCoy rings, we introduce the concept ofα-skewπ-McCoy rings, and we study the relationships with another two new generalizations,α-skewπ1-McCoy rings andα-skewπ2-McCoy rings, observing the relations withα-skew McCoy rings,π-McCoy rings,α-skew Armendariz rings,π-regular rings, and other kinds of rings. Also, we investigate conditions such thatα-skewπ1-McCoy rings implyα-skewπ-McCoy rings andα-skewπ2-McCoy rings. We show that in the case whereRis a nonreduced ring, ifRis 2-primal, thenRis anα-skewπ-McCoy ring. And, letRbe a weak (α,δ)-compatible ring; ifRis anα-skewπ1-McCoy ring, thenRisα-skewπ2-McCoy.

A note on complete rings of quotients and McCoy rings

If a (commutative unital) ring \(A\) is reduced and coincides with its total quotient ring, then \(A\) satisfies Property A (that is, \(A\) is a McCoy ring) if and only if the inclusion of \(A\) in its complete ring of quotients \(C(A)\) is a survival extension. The “if” assertion fails if one deletes the hypothesis that \(A\) is reduced. This is shown by using the idealization construction to construct a suitable ring \(A\) and then identifying its complete ring of quotients (which turns out to be a related idealization). Related characterizations of von Neumann regular rings are also given with the aid of the going-down property GD of ring extensions. For instance, a ring \(A\) is von Neumann regular if and only if \(A\) is a reduced McCoy ring that coincides with its total quotient ring such that \(A \subseteq C(A)\) satisfies GD.

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.

On Rings Having McCoy-Like Conditions

In [41], Nielsen proves that all reversible rings are McCoy and gives an example of a semicommutative ring that is not right McCoy. At the same time, he also shows that semicommutative rings do have a property close to the McCoy condition. In this article we study weak McCoy rings as a common generalization of McCoy rings and weak Armendariz rings. Relations between the weak McCoy property and other standard ring theoretic properties is considered. We also study the weak skew McCoy condition, a generalization of the standard weak McCoy condition from polynomials to skew polynomial rings. We resolve the structure of weak skew McCoy rings and obtain various necessary or sufficient conditions for a ring to be weak skew McCoy, unifying and generalizing a number of known McCoy-like conditions in the special cases. Constructing various examples, we classify how the weak McCoy property behaves under various ring extensions. As a consequence we extend and unify several known results related to McCoy rings and Armendariz rings [6, 35, 38, 43, 49].

On \alpha-skew McCoy modules

Let \alpha be a ring endomorphism. Extending the notions of McCoy modules and \alpha-skew McCoy rings, we introduce the notion of \alpha-skew McCoy modules, which can also be regarded as a generalization of \alpha-skew Armendariz modules. A number of illustrative examples are given. Various properties of these modules are developed, and equivalent conditions for \alpha-skew McCoy modules are established. Furthermore, we study the relationship between a module and its polynomial module.

Power-serieswise McCoy Rings

In this paper, we introduce power-serieswise McCoy rings, which are a generalization of power-serieswise Armendariz rings, and investigate their properties. We show that a ring R is power-serieswise McCoy if and only if the ring consisting of n × n upper triangular matrices with equal diagonal entries over R is power-serieswise McCoy. We also prove that a direct product of rings is power-serieswise McCoy if and only if each of its factors is power-serieswise McCoy. Meanwhile we show that power-serieswise McCoy rings may be neither semi-commutative nor power-serieswise Armendariz.

On McCoy modules

Extending the notion of McCoy rings, we introduce the class of McCoy modules. Over a given ring R, it contains the class of Armendariz modules (over R). Some properties of this class of modules are established, and equivalent conditions for McCoy modules are given. Moreover, we study the relationship between a module and its polynomial module. Several known results relating to McCoy rings can be obtained as corollaries of our results.

ON A GENERALIZATION OF THE MCCOY CONDITION

We in this note consider a new concept, so called <TEX>$\pi$</TEX>-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of <TEX>$\pi$</TEX>-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of <TEX>$\pi$</TEX>-McCoy rings, observing the relations among <TEX>$\pi$</TEX>-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and (<TEX>$\pi-$</TEX>)regular rings. It is proved that the n by n full matrix rings (<TEX>$n\geq2$</TEX>) over reduced rings are not <TEX>$\pi$</TEX>-McCoy, finding <TEX>$\pi$</TEX>-McCoy matrix rings over non-reduced rings. It is shown that the <TEX>$\pi$</TEX>-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of <TEX>$\pi$</TEX>-McCoy rings are also examined.

McCoy Rings Relative to a Monoid

For a monoid M, we introduce M-McCoy rings, which are a generalization of McCoy rings and M-Armendariz rings; and investigate their properties. We first show that all reversible rings are right M-McCoy, where M is a u.p.-monoid. We also show that all right duo rings are right M-McCoy, where M is a strictly totally ordered monoid. Then we show that semicommutative rings and 2-primal rings do have a property close to the M-McCoy condition. Moreover, it is shown that a finitely generated Abelian group G is torsion free if and only if there exists a ring R such that R is G-McCoy. Consequently, several known results on right McCoy rings are extended to a general setting.

The McCoy Condition on Skew Polynomial Rings

Based on a theorem of McCoy on commutative rings, Nielsen called a ring R right McCoy if, for any nonzero polynomials f(x), g(x) over R, f(x)g(x) = 0 implies f(x)r = 0 for some 0 ≠ r ∈ R. In this note, we consider a skew version of these rings, called σ-skew McCoy rings, with respect to a ring endomorphism σ. When σ is the identity endomorphism, this coincides with the notion of a right McCoy ring. Basic properties of σ-skew McCoy rings are observed, and some of the known results on right McCoy rings are obtained as corollaries.

Extensions of McCoy Rings

A ring R is called right McCoy if whenever polynomials f(x), g(x) ∈ R[x]∖{0} satisfy f(x)g(x)=0, there exists a nonzero r ∈ R such that f(x)r=0. We continue in this paper the study of right McCoy rings by Nielsen [8]. We first consider properties and basic extensions of right McCoy rings, providing many examples in the process. Next, we show that if R is a right McCoy ring, then R[x] and R[x]/(xn) are right McCoy rings, where (xn) is the ideal generated by xn and n is a positive integer; and that for a right Ore ring R with Q its classical right quotient ring, R is right McCoy if and only if Q is right McCoy.

Extensions of Rings Having McCoy Condition

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

McCoy rings and zero-divisors

We investigate relations between the McCoy property and other standard ring theoretic properties. For example, we prove that the McCoy property does not pass to power series rings. We also classify how the McCoy property behaves under direct products and direct sums. We prove that McCoy rings with 1 are Dedekind finite, but not necessarily Abelian. In the other direction, we prove that duo rings, and many semi-commutative rings, are McCoy. Degree variations are defined, studied, and classified. The McCoy property is shown to behave poorly with respect to Morita equivalence and (infinite) matrix constructions.

A question on McCoy rings

Nelsen [J. Algebra 298 (2006) 134–141] asked whether there is a natural class of McCoy rings which includes all reversible rings and all rings R such that R[X] is semi-commutative. In this paper, some new equivalent conditions of McCoy rings are given. One of them is used to answer this question in the affirmative. Finally, an example is given which is McCoy and semi-commutative, but it is not reversible and does not have the property that R[x] is semi-commutative.