On extensions of Baer and quasi-Baer modules

Let R be a ring, MR a module, S a monoid, ω : S → End(R) a monoid homomorphism and R * S a skew monoid ring. Then M[S] = {m1g1 + · · · + mngn | n ≥ 1, mi ∈ M and gi ∈ S for each 1 ≤ i ≤ n} is a module over R ∗ S. A module MR is Baer (resp. quasi-Baer) if the annihilator of every subset (resp. submodule) of M is generated by an idempotent of R. In this paper we impose S-compatibility assumption on the module MR and prove: (1) MR is quasi-Baer if and only if M[s]R∗S is quasi-Baer, (2) MR is Baer (resp. p.p) if and only if M[S]R∗S is Baer (resp. p.p), where MR is S-skew Armendariz, (3) MR satisfies the ascending chain condition on annihilator of submodules if and only if so does M[S]R∗S, where MR is S-skew quasi-Armendariz.

Idempotent elements and uniquely clean property of skew monoid rings

Let R be a ring with an endomorphism σ and F ∪ {0} the free monoid generated by U = {u1, ..., ut} with 0 added, and M = F ∪ {0}/(I) where I is the set of certain monomial in U such that Mn = 0, for some n. Then we can form the non-semiprime skew monoid ring R[M; σ]. An element a ∈ R is uniquely strongly clean if a has a unique expression as a = e + u, where e is an idempotent and u is a unit with ea = ae. We show that a σ-compatible ring R is uniquely clean if and only if R[M; σ] is a uniquely clean ring. If R is strongly π-regular and uniquely strongly clean, then R[M; σ] is uniquely strongly clean. It is also shown that idempotents of R[M; σ] (and hence the ring R[x; σ]=(xn)) are conjugate to idempotents of R and we apply this to show that R[M; σ] over a projective-free ring R is projective-free. It is also proved that if R is semi-abelian and σ(e) = e for each idempotent e ∈ R, then R[M; σ] is a semi-abelian ring.

The McCoy condition on skew monoid rings

Let [Formula: see text] be an associative ring with identity, [Formula: see text] a monoid and [Formula: see text] a monoid homomorphism. When [Formula: see text] is a u.p.-monoid and [Formula: see text] is a reversible [Formula: see text]-compatible ring, then we observe that [Formula: see text] satisfies a McCoy-type property, in the context of skew monoid ring [Formula: see text]. We introduce and study the [Formula: see text]-McCoy condition on [Formula: see text], a generalization of the standard McCoy condition from polynomial rings to skew monoid rings. Several examples of reversible [Formula: see text]-compatible rings and also various examples of [Formula: see text]-McCoy rings are provided. As an application of [Formula: see text]-McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew monoid ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text].

Answers to Some Questions Concerning Rings with Property (A)

A ring R has right property (A) whenever a finitely generated two-sided ideal of R consisting entirely of left zero-divisors has a non-zero right annihilator. As the main result of this paper we give answers to two questions related to property (A), raised by Hong et al. One of the questions has a positive answer and we obtain it as a simple conclusion of the fact that if R is a right duo ring and M is a u.p.-monoid (unique product monoid), then R is right M-McCoy and the monoid ring R[M] has right property (A). The second question has a negative answer and we demonstrate this by constructing a suitable example.

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.

Skew monoid rings with annihilator conditions

One of the most active and important research areas in noncommutative algebra is the investigation of skew monoid rings. Given a ring [Formula: see text] and a monoid [Formula: see text], we study the structure of the set of zero divisors and nilpotent elements in skew monoid ring [Formula: see text]. In the process we introduce a nil analog of the [Formula: see text]-skew [Formula: see text]-McCoy ring defined by Alhevaz and Kiani in [McCoy property of skew Laurent polynomials and power series rings, J. Algebra Appl. 13(2) (2014), Article ID: 1350083, 23pp.] and introduce the concept of so-called [Formula: see text]-skew nil [Formula: see text]-McCoy ring, which is a common generalization of [Formula: see text]-skew [Formula: see text]-McCoy rings, nil-McCoy rings and McCoy rings relative to a monoid. It is done by considering the nil-McCoy condition on a skew monoid ring [Formula: see text] instead of the polynomial ring [Formula: see text]. We also obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-skew nil [Formula: see text]-McCoy. Among other results, we prove that each regular [Formula: see text]-skew [Formula: see text]-McCoy ring [Formula: see text] is abelian (i.e. idempotents are central), where [Formula: see text] is any monoid with an element of infinite order and [Formula: see text] is a compatible monoid homomorphism. This answers, in a much more general setting, a question posed in [A. R. Nasr-Isfahani, On semiprime right Goldie McCoy rings, Comm. Algebra 42(4) (2014) 1565–1570], in the positive. Furthermore, we provide various examples and classify how the nil [Formula: see text]-McCoy rings behaves under various ring extensions.

On the Grothendieck ring of varieties

Let K0(Vark) denote the Grothendieck ring of k-varieties over an algebraically closed field k. Larsen and Lunts asked if two k-varieties having the same class in K0(Vark) are piecewise isomorphic. Gromov asked if a birational self-map of a k-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb{Z}$[$\mathfrak{B}$] where $\mathfrak{B}$ denotes the multiplicative monoid of birational equivalence classes of irreducible k-varieties.

A new class of non-semiprime quasi-Armendariz rings

Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168 (2002), 45–52] studied relations between the set of annihilators in a ring R and the set of annihilators in a polynomial extension R[x] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring R and a monoid M such that the monoid ring R[M] is quasi-Armendariz. As a consequence we show that if R is a right APP-ring, then R[x]=(xn) and hence the trivial extension T(R,R) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.