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In this work, we first prove a necessary and sufficient condition for a pairs of linear codes over finite rings to be linear complementary pairs (abbreviated to LCPs). In particular, a judging criterion of free LCP of codes over finite commutative rings is obtained. Using the criterion of free LCP of codes, we construct a maximum-distance-separable (MDS) LCP of codes over ring $$\mathbb {Z}_4$$ . Then, all the possible LCP of codes over chain rings are determined. We also generalize the criterions for constacyclic and quasi-cyclic LCP of codes over finite fields to constacyclic and quasi-cyclic LCP of codes over chain rings. Finally, we give a characterization of LCP of codes over principal ideal rings. Under suitable conditions, we also obtain the judging criterion for a pairs of cyclic codes over principal ideal rings $$\mathbb {Z}_{k}$$ to be LCP, and illustrate a MDS LCP of cyclic codes over the principal ideal ring $$\mathbb {Z}_{15}$$ .

Definirali bomo relacijo leve ekvivalence na matrikah, katerih elementi pripadajo nekemu glavnemu kolobarju. Poiskali bomo Hermitovo normalno formo in obravnavali njene posledice. We will define the left equivalence relation on matrices whose elements belong to a principal ideal ring. We will find the Hermite normal form and explore some of its consequences.

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we show that any finite-dimensional indecomposable [Formula: see text]-module is generated by one element. In particular, any indecomposable submodule of [Formula: see text] under the adjoint action is generated by a special element of [Formula: see text]. Using this result, we show that the Hopf algebra [Formula: see text] is a principal ideal ring, i.e., any two-sided ideal of [Formula: see text] is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras.

Let Hm,n be the mn2-dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. In this paper, we show that any finite dimensional indecomposable Hm,n-module is a cy...

In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor...

Several different versions of “factoriality” have been defined for commutative rings with zero divisors. We apply semigroup theory to study these notions in the context of a commutative monoid ring R[S], determining necessary and sufficient conditions for R[S] to be various kinds of “unique factorization rings.” Our work generalizes Anderson et al.’s results about “unique factorization” in R[X], Gilmer and Parker’s characterization of factorial monoid domains, and Hardy and Shores’s classification of when R[S] is a principal ideal ring (for S cancellative). Along the way, we determine when R[S] is “restricted cancellative” or satisfies various “(restricted) ideal cancellation laws.”

In this paper we present new techniques for improving the computation of strong Grobner bases over a principal ideal ring R . More precisely, we describe how to lift a strong Grobner basis along a canonical projection R → R / n , n ≠ 0 , and along a ring isomorphism R → R 1 × R 2 . We then apply this to the computation of strong Grobner bases over a non-trivial quotient of a principal ideal domain R / n R . The idea is to run a standard Grobner basis algorithm pretending R / n R to be field. If we discover a non-invertible leading coefficient c, we use this information to try to split n = a b with coprime a , b . If this is possible, we recursively reduce the original computation to two strong Grobner bases computations over R / a R and R / b R respectively. If no such c is discovered, the returned Grobner basis is already a strong Grobner basis for the input ideal over R / n R .

Let $$(R, \pi R, k,e)$$ be a commutative special principal ideal ring (SPIR), where R is its maximal ideal, k its residual field and e the index of nilpotency of $$\pi$$ . An ideal I of R[X] is called an integral ideal if it contains a monic polynomial. In this paper, we show that if R is a SPIR, then an ideal I in R[X] is integral if and only if $${\overline{I}} \ne {{\overline{0}}}$$ in k[X]. Furthermore, the lowest degree of monic polynomials in I is exactly the lowest degree of nonzero polynomials in $${\overline{I}}$$ .

Let p be an odd prime, s and m be positive integers. Cyclic codes of length $2p^{{s}}$ over $\mathbb {F}_{{p}^{{m}}}$ are the ideals $\langle ( {x}-1)^{i}({x}+1)^{j} \rangle $ , where $0 \le {i}, {j} \le {p}^{ {s}}$ , of the principal ideal ring $\mathbb {F}_{{p}^{{m}}}[{x}]/\langle {x}^{2\textit {p}^{\textit {s}}}-1 \rangle $ . Using this structure, the symbol-pair distances of all cyclic codes of length $2\textit {p}^{\textit {s}}$ over $\mathbb F_{{p}^{{m}}}$ are completely determined. In addition, we establish all MDS symbol-pair cyclic codes of length $2\textit {p}^{\textit {s}}$ over $\mathbb F_{{p}^{{m}}}$ . Some MDS symbol-pair cyclic codes are better than all the known ones. Among others, we discuss possible applications to construct quantum MDS symbol-pair codes.

The rings considered in this article are commutative with identity. For an ideal $I$ of a ring $R$, we denote the annihilator of $I$ in $R$ by $Ann(I)$. An ideal $I$ of a ring $R$ is said to be an exact annihilating ideal if there exists a non-zero ideal $J$ of $R$ such that $Ann(I) = J$ and $Ann(J) = I$. For a ring $R$, we denote the set of all exact annihilating ideals of $R$ by $\mathbb{EA}(R)$ and $\mathbb{EA}(R)\backslash \{(0)\}$ by $\mathbb{EA}(R)^{*}$. Let $R$ be a ring such that $\mathbb{EA}(R)^{*}\neq \emptyset$. With $R$, in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of $R$, denoted by $\mathbb{EAG}(R)$ whose vertex set is $\mathbb{EA}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent if and only if $Ann(I) = J$ and $Ann(J) = I$. In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of $\mathbb{EAG}(R)$, where either $R$ is a special principal ideal ring or $R$ is a reduced ring which admits only a finite number of minimal prime ideals.

We carry out a study of rings $R$ for which $\operatorname{Hom}_R(M,N)\neq 0$ for all nonzero $ N\leq M_R$. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Köthe ring $R$ is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when $R$ is retractable.

Let K be the quotient field of a principal ideal ring R, and let σ = (σij) be a full (respectively, elementary) net of order n ≥ 2 (respectively, n ≥ 3) over K such that the additive subgroups σij are nonzero R-modules. It is proved that, up to conjugation by a diagonal matrix, all σij are ideals of a fixed intermediate subring P, R ⊆ P ⊆ K.

Let D 2 n = 〈 x , y ∣ x n = 1 , y 2 = 1 , y x y = x − 1 〉 be a dihedral group, and R = GR ( p 2 , m ) be a Galois ring of characteristic p 2 and cardinality p 2 m where p is a prime. Left ideals of the group ring R [ D 2 n ] are called left dihedral codes over R of length 2 n , and abbreviated as left D 2 n -codes over R . Let gcd ( n , p ) = 1 in this paper. Then any left D 2 n -code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A i and outer codes C i , where A i is a cyclic code over R of length n and C i is a skew cyclic code of length 2 over a Galois ring or principal ideal ring extension of R . Specifically, a generator matrix and basic parameters for each outer code C i are given. A formula to count the number of these codes is obtained and the dual code for each left D 2 n -code is determined. Moreover, all self-dual left D 2 n -codes and self-orthogonal left D 2 n -codes over R are presented.

Because of its interesting applications in coding theory, cryptography, and algebraic combinatorics, in recent decades a lot of attention has been paid to the algebraic structure of the ring of polynomials R[x], where R is a finite commutative ring with identity. Motivated by this popularity, in this paper we determine when R[x] is a principal ideal ring. In fact, we prove that R[x] is a principal ideal ring if and only if R is a finite direct product of finite fields. MSC2010: 13F10, 13F20, 16P10, 13C05. Resumen Debido a sus interesantes aplicaciones en teoría de códigos, criptografía y combinatoria algebraica, en décadas recientes se ha incrementado la atención en la estructura algebraica del anillo de polinomios R[x], donde R es un anillo conmutativo finito con identidad. Motivados por esta popularidad, en este artículo determinamos cuándo R[x] es un anillo de ideales principales. De hecho, demostramos que R[x] es un anillo de ideales principales, si y sólo si, R es un producto directo finito de campos finitos.

Let R be a (possibly noncommutative) finite principal ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module $$R^n$$ is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combinations with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull of the vectors selected so far. The output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. In this paper we investigate the properties of such lexicodes over finite principal ideal rings and show that the total ordering of the ring elements has to respect containment of ideals for the algorithm to produce meaningful results. Only then it is guaranteed that the algorithm is exhaustive and thus produces codes that are maximal with respect to inclusion. It is further illustrated that the output of the algorithm heavily depends on the total ordering and chosen basis.

In this work, we study the structure of multivariable modular codes over finite chain rings when the ambient space is a principal ideal ring. We also provide some applications to additive modular codes over the finite field F 4 .

In this paper, we consider the ideal based zero divisor graph ΓI(R) of a commutative ring R. We discuss some graph theoretical properties of ΓI(R) in relation with zero divisor graph. We also relate certain parameters like vertex chromatic number, maximum degree and minimum degree for the graph ΓI(R) with that of Γ(R/I ). Further we determine a necessary and sufficient condition for the graph to be Eulerian and regular.

We study the indecomposability and isomorphism of objects from the category of monomial matrices Mmat(K) over a commutative local principal ideal ring K (whose objects are square monomial matrices and the morphisms from X to Y are matrices C such that XC = CY). We also study the subcategory Mmat0(K) of the category Mmat(K) with the same objects and solely with morphisms that are monomial matrices.

In this paper, the determinants of n × n matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of n × n matrices over a commutative finite chain ring R of a fixed determinant a is determined for all a ∈ R and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of n × n matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.

The Nagata ring R(X) and the Serre’s conjecture ring R⟨X⟩ are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this pa...

The study of the class of van der Waerden rings was initiated by Comfort, Remus and Szambien in [7] . A compact ring ( R , T ) is a van der Waerden ring if and only if T is the only totally bounded ring topology on R . In this paper the study of this class of compact rings is continued. We prove that an Abelian compact van der Waerden ring R has the form R = ( ∏ i ∈ I R i ) × L , where L is a compact radical ring with radical topology and R i ( i ∈ I ) is a compact local topologically finitely generated ring. It is shown that every van der Waerden ring is totally disconnected and metrizable. Furthermore, the class of van der Waerden rings is closed under extensions. Different classes of compact rings are studied which are closely related to van der Waerden rings.

An interesting result, obtaining by some theorems of Asano, Kothe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ℳ of the form ℳ = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (res...

We prove that for any valuation ring R of Krull dimension ≤1 or any commutative local ring R with Krull dimension 0 and n≥3, the special linear group SLn(R[x]) coincides with the group of elementary matrices En(R[x]), and for an arbitrary arithmetical ring R of Krull dimension ≤1 and n≥3, the group SLn(R[x])=SLn(R)⋅En(R[x]). These give partial solutions to two conjectures of Yengui.

Let R be a ring and $$\delta $$ is a derivation of R. In this paper, it is proved that, under suitable conditions, the differential polynomial ring $$R[x;\delta ]$$ has the same triangulating dimension as R. Furthermore, for a piecewise prime ring, we determine a large class of the differential polynomial ring which have a generalized triangular matrix representation for which the diagonal rings are prime.

We prove that a ring which is a sum of two PI subrings is a PI ring as well. This answers a question of K. I. Beidar and A. V. Mikhalev that has been open since 1995.

Let R be a semiprime ring with center Z(R) and with extended centroid C. Suppose that τ:R→R is an anti-homomorphism such that the image of τ has small centralizer. It is proved that the fol...

Let R be a commutative ring with identity and let Nil ( R ) be the ideal of all nilpotent elements of R . Let I ( R ) = { I : I is a non-trivial ideal of R and there exists a non-trivial ideal J such that I J ⊆ Nil ( R ) } . The nil-graph of ideals of R is defined as the simple undirected graph A G N ( R ) whose vertex set is I ( R ) and two distinct vertices I and J are adjacent if and only if I J ⊆ Nil ( R ) . In this paper, we study the planarity and genus of A G N ( R ) . In particular, we have characterized all commutative Artin rings R for which the genus of A G N ( R ) is either zero or one.

In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation.

Let R be a finite commutative ring with identity. In this paper, given an integer $$k\ge 2$$ , we obtain an exact formula for the number of ways to represent each element of R as the sum of k exceptional units. This generalizes a recent result of Quan-Hui Yang and Qing-Qing Zhao for the case where R is the ring $${\mathbb {Z}}_n$$ of residue classes modulo n.

Entangling properties of a mixed two-qubit system can be described by local homogeneous unitary invariant polynomials in the elements of the density matrix. The structure of the corresponding ring of invariant polynomials for a special subclass of states, the so-called mixed X-states, is established. It is shown that for the X-states there is an injective ring homomorphism of the quotient ring of SU(2)×SU(2)-invariant polynomials modulo its syzygy ideal to the SO(2) × SO(2)-invariant ring freely generated by five homogeneous polynomials of degrees 1, 1, 1, 2, 2.

A result of Nakayama and Skornyakov states that a ring R is an Artinian serial ring if and only if every R-module is serial. This motivated us to study commutative rings for which every proper ideal is serial. In this paper, we determine completely the structure of commutative rings R of which every proper ideal is serial. It is shown that every proper ideal of R is serial, if and only if, either R is a serial ring, or R is a local ring with maximal ideal ${\mathcal {M}}$ such that there exist a uniserial module U and a semisimple module T with ${\mathcal {M}}=U\oplus T$ . Moreover, in the latter case, every proper ideal of R is isomorphic to $U^{\prime }\oplus T^{\prime }$ , for some $U^{\prime }\leq U$ and $T^{\prime }\leq T$ . Furthermore, it is shown that every proper ideal of a commutative Noetherian ring R is serial, if and only if, either R is a finite direct product of discrete valuation domains and local Artinian principal ideal rings, or R is a local ring with maximal ideal ${\mathcal {M}}$ containing a set of elements {w 1,…,w n } such that ${\mathcal {M}}=\bigoplus _{i=1}^{n} Rw_{i}$ with at most one non-simple summand. Moreover, another equivalent condition states that: there exists an integer n ≥ 1 such that every proper ideal of R is a direct sum of at most n uniserial R-modules. Finally, we discuss some examples to illustrate our results.

A ring R is said to be left pure-hereditary (resp. RD-hereditary) if every left ideal of R is pure-projective (resp. RD-projective). In this paper, some properties and examples of these rings, which are nontrivial generalizations of hereditary rings, are given. For instance, we show that if R is a left RD-hereditary left nonsingular ring, then R is left Noetherian if and only if u . dim ( R R ) ∞ . Also, we show that a ring R is quasi-Frobenius if and only if R is a left FGF, left coherent right pure-injective ring. A ring R is said to be left FP-hereditary if every left ideal of R is FP-projective. It is shown that if R is a left CF ring, then R is left Noetherian if and only if R is left pure-hereditary, if and only if R is left FP-hereditary, if and only if R is left coherent. It is shown that every left self-injective left FP-hereditary ring is semiperfect. Finally, it is shown that a ring R is left FP-hereditary (resp. left coherent) if and only if every submodule (resp. finitely generated submodule) of a projective left R-module is FP-projective, if and only if every pure factor module of an injective left R-module is injective (resp. FP-injective), if and only if for each FP-injective left R-module U, E ( U ) / U is injective (resp. FP-injective).

A commutative ring R is an elementary divisor ring if every matrix over R admits a diagonal reduction. In this paper, we define the term ‘Zabvasky subset’ of a ring to study diagonal matrix reduction. Let S be a Zabavsky subset of a Hermite ring R. We prove that R is an elementary divisor ring if and only if with implies that there exist such that . If with implies that there exists a such that , then R is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

A well-known theorem by S. A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x;D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x;D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p > 0. The main idea of the proof introduces a new way of embedding differential polynomial rings into bigger rings, which we name platinum rings, plus a key part of the proof involves the solution of matrix theory-based problems.

A ring R is called a right PS-ring if its socle, Soc(R R ), is projective. Nicholson and Watters have shown that if R is a right PS-ring, then so are the polynomial ring R[x] and power series ring R[[x]]. In this paper, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does the skew inverse power series ring R[[x −1; α, δ]] and the skew polynomial ring R[x; α, δ], where R is an associative ring equipped with an automorphism α and an α-derivation δ. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.

The reversible property is an important role in noncommutative ring theory. Recently, the study of the reversible ring property on nilpotent elements is established by Abdul-Jabbar et al., introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We here study this property skewed by a ring endomorphism α, and such ring is called a right α-skew CNZ ring which is an extension of CNZ rings as well as a generalization of right α-skew reversible rings, and then investigate the structure of right α-skew CNZ rings and their related properties. Consequently, several known results are obtained as corollaries of our results.

We consider the problem of describing the form of biderivations of a triangular ring. Our approach is based on the notion of the maximal left ring of quotients, which enables us to generalize Benkovic’s result on biderivations (Benkovic in Linear Algebra Appl 431:1587–1602, 2009). Our result is applied to block upper triangular matrix rings and nest algebras.

Let R be a simple GPI-ring (i.e., a simple ring satisfying a nontrivial generalized polynomial identity) with a nonzero derivation delta and with Martindale symmetric ring of quotients Q. Motivated by the Noether-Skolem theorem, we characterize linear differential maps phi: x bar right arrow Sigma(i,j) a(ij)delta(j) (X) b(ij) for x is an element of R, where a(ij),b(ij) are finitely many elements in Q, such that phi(R) subset of [R,R]. The result is described and proved in terms of polynomials in Q[t;delta], the Ore extension of Q by the derivation delta.

Let R be a ring with identity. Then {0} and R are the only additive subgroups of R if and only if R is isomorphic (as a ring with identity) to (exactly) one of {0}, Z/pZ for a prime number p. Also, each additive subgroup of R is a one-sided ideal of R if and only if R is isomorphic to (exactly) one of {0}, Z, Z/nZ for an integer n ≥ 2. This note could find classroom use in a first course on abstract algebra as enrichment material for the unit on ring theory.

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a1⋯am∈I for a1,…, am∈R∖U(R), then there are n of the ai’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bezout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m − 1 maximal ideals.

In this paper, we concern the Principal Ideal Theorem (PIT) for w-Noetherian rings. Let R be a w-Noetherian ring and a be a nonzero nonunit element of R. If p is a prime ideal of R minimal over (a), then ht p ≦ 1.

Let K be an associative and commutative ring with 1, and k a subring of K such that and the elements 2, 3 are invertible in k. Let n be an integer and an invertible diagonal n by n matrix over k. In the paper, Lie subrings of the general linear Lie ring containing the special orthogonal Lie ring are described provided that .

For a finite dimensional division ring D we establish a condition for the connectedness of the commuting graph Γ ( M n ( D ) ) . Furthermore, if the graph Γ ( M n ( D ) ) is connected, we prove that its diameter cannot be greater that six.

This paper concerns mainly on various ring properties of some subrings of a skew triangular matrix ring. Necessary and sufficient conditions are obtained for a skew triangular matrix ring with constant diagonal over a ring to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, clean, (weakly) nil clean, exchange, uniquely (nil) clean, Hermite ring, stably finite, semiregular and I-ring. It is also proved that the projective-free property of a ring preserves by a skew triangular matrix ring with constant diagonal.

The self-idealization of a commutative ring R is isomorphic to the ring \(R[x]/(x^2)\) or, equivalently, the ring of upper-triangular Toeplitz matrices \({{\mathrm{\mathcal {T}}}}(R)=\left\{ \left( \begin{matrix} a &{} b \\ 0 &{} a \end{matrix}\right) :a,b\in R\right\} \). Recently, Chang and Smertnig characterized the sets of lengths of factorizations in \({{\mathrm{\mathcal {T}}}}(D)\) where D is a principal ideal domain. In this work, in addition to correcting an error in their paper, we extend the study to \({{\mathrm{\mathcal {T}}}}(R)\) when R is either a principal ideal ring or a unique factorization ring.

We say that a class 𝒫 of right modules over a fixed ring R is an epic class if it is closed under homomorphic images. For an arbitrary epic class 𝒫, we define a 𝒫-dimension of modules that measures how far modules are from the modules in the class 𝒫. For an epic class 𝒫 consisting of indecomposable modules, first we characterize rings whose modules have 𝒫-dimension. In fact, we show that every right R-module has 𝒫-dimension if and only if R is a semisimple Artinan ring. Then we study fully Hopfian modules with 𝒫-dimension. In particular, we show that a commutative ring R with 𝒫-dimension (resp. finite 𝒫-dimension) is either local or Noetherian (resp. Artinian). Finally, we show that Matm(R) is a right Kothe ring for some m if and only if every (left) right module is a direct sum of modules of 𝒫-dimension at most n for some n, if and only if R is a pure semisimple ring.

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.

Let R be a ring equipped with an automorphism α and an α-derivation δ. We studied on the relationship between the quasi Baerness and (α, δ)-quasi Baerness of a ring R and these of the inverse skew Laurent series ring R((x−1; α, δ)), in case R is an (α, δ)-weakly rigid ring. Also we proved that for a semicommutative (α, δ)-weakly rigid ring R, R is Baer if and only if so is R((x−1; α, δ)). Moreover for an (α, δ)-weakly rigid ring R, it is shown that the inverse skew Laurent series ring R((x−1; α, δ)) is left p.q.-Baer if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.

Let R be a ring with an automorphism α and a derivation δ. In this article we provide necessary and sufficient conditions for a skew polynomial ring R[x;α] and differential polynomial ring R[x;δ] to be 2-primal. We compute the Jacobson radical and the set of unit elements of a 2-primal skew polynomial ring R[x;α] and differential polynomial ring R[x;δ]. Also we establish the lower bounds on the stable range of a 2-primal skew polynomial ring R[x;α] and differential polynomial ring R[x;δ]. As an application we show that if R is 2-primal then the nth Weyl algebra over R is 2-primal and in this case J(An(R))=An(Nil∗(R)). As a consequence, we extend and unify several known results of [4], [8], [10], [18], [19], and [22].