Simplicity of iterated Ore extensions

Here we study the simplicity of an iterated Ore extension of a unital ring [Formula: see text]. We give necessary conditions for the simplicity of an iterated Ore extension when [Formula: see text] is a commutative domain. A class of iterated Ore extensions, namely the differential polynomial ring [Formula: see text] in [Formula: see text]-variables is considered. The conditions for a commutative domain [Formula: see text] of characteristic zero to be a maximal commutative subring of its differential polynomial ring [Formula: see text] are given, and the necessary and sufficient conditions for [Formula: see text] to be simple are also found.

PRINC domains and comaximal factorization domains

The notion of a PRINC domain was introduced by Salce and Zanardo [Products of elementary and idempotent matrices over integral domains, Linear Algebra Appl. 452 (2014) 130–152], motivated by the investigation of the products of idempotent matrices with entries in a commutative domain. An integral domain [Formula: see text] is a PRINC domain if every two-generated invertible ideal of [Formula: see text] is principal. PRINC domains are closely related to the notion of a unique comaximal factorization domain, introduced by McAdam and Swan [Unique comaximal factorization, J. Algebra 276 (2004) 180–192]. In this paper, we prove that there exist large classes of PRINC domains which are not comaximal factorization domains, using diverse kinds of constructions. We also produce PRINC domains that are neither comaximal factorization domains nor projective-free.

$\omega$-Euclidean domain and Laurent series

It is proved that a commutative domain $R$ is $\omega$-Euclidean if and only if the ring of formal Laurent series over $R$ is $\omega$-Euclidean domain. It is also proved that every singular matrice over ring of formal Laurent series $R_{X}$ are products of idempotent matrices if $R$ is $\omega$-Euclidean domain.

Effective results for unit points on curves over finitely generated domains

Let A be a commutative domain of characteristic 0 which is finitely generated over ℤ as a ℤ-algebra. Denote by A* the unit group of A and by K the algebraic closure of the quotient field K of A. We shall prove effective finiteness results for the elements of the set \begin{equation*} \mathcal{C}:=\{ (x,y)\in (A^*)^2 | F(x,y)=0 \} \end{equation*} where F(X, Y) is a non-constant polynomial with coefficients in A which is not divisible over K by any polynomial of the form XmYn - α or Xm - α Yn, with m, n ∈ ℤ⩾0, max(m, n) > 0, α ∈ K*. This result is a common generalisation of effective results of Evertse and Győry [12] on S-unit equations over finitely generated domains, of Bombieri and Gubler [5] on the equation F(x, y) = 0 over S-units of number fields, and it is an effective version of Lang's general but ineffective theorem [20] on this equation over finitely generated domains. The conditions that A is finitely generated and F is not divisible by any polynomial of the above type are essentially necessary.

A NOTE ON DENSITY AND THE DIRICHLET CONDITION

Motivated by topological approaches to Euclid and Dirichlet's theorems on infinitude of primes, we introduce and study [Formula: see text]-coprime topologies on a commutative ring R with an identity and without zero divisors. For infinite semiprimitive commutative domain R of finite character (i.e. every nonzero element of R is contained in at most finitely many maximal ideals of R), we characterize its subsets A for which the Dirichlet condition, requiring the existence of infinitely many pairwise nonassociated elements from A in every open set in the invertible topology, is satisfied.

PROPERTIES OF K-RINGS AND RINGS SATISFYING SIMILAR CONDITIONS

Jacobson introduced the concept of K-rings, continuing the investigation of Kaplansky and Herstein into the commutativity of rings. In this note we focus on the ring-theoretic properties of K-rings. We first construct basic examples of K-rings to be handled easily. It is shown that a semiprime K-ring of bounded index of nilpotency is a commutative domain. It is proved that if R is a prime K-ring then its classical quotient ring is a local ring with a nil Jacobson radical. We also show that if R is a π-regular K-ring then R/P is a field for every strongly prime ideal P of R. The basic structure of a condition, unifying K-rings and reversible rings, is studied with respect to zero-divisors in matrices and polynomials.

DIVISIBLE ENVELOPES, $\mathcal{P}_1$-COVERS AND WEAK-INJECTIVE MODULES

We consider the problem of characterizing the commutative domains such that every module admits a divisible envelope or a [Formula: see text]-cover, where [Formula: see text] is the class of modules of projective dimension at most one. We prove that each one of the two conditions is satisfied exactly for the class of almost perfect domains, that is commutative domains such that all of their proper quotients are perfect rings. Moreover, we show that the class of weak-injective modules over a commutative domain is closed under direct sums if and only if the domain is almost perfect.

A GENERALIZATION OF BAER'S LOWER NILRADICAL FOR MODULES

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals [Formula: see text] and for all submodules N ⊆ M, [Formula: see text] implies that [Formula: see text] or [Formula: see text]. We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.rad R(M)] and Baer's lower nilradical for a module [denoted by Nil *(RM)]. For a module RM, cl.rad R(M) is defined to be the intersection of all classical prime submodules of M and Nil *(RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.rad R(M) = Nil *(RM) and, for any module M over a left Artinian ring R, cl.rad R(M) = Nil *(RM) = Rad (M) = Jac (R)M. In particular, if R is a commutative Noetherian domain with dim (R) ≤ 1, then for any module M, we have cl.rad R(M) = Nil *(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim (R) ≤ 1 (or over a commutative domain R with dim (R) ≤ 1), every semiprime submodule of any module is an intersection of classical prime submodules.