Semilocal rings with <Emphasis Type="Italic">n</Emphasis>-Engel multiplicative group

An associative ring R with unit element is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that the multiplicative group R* of a semilocal ring R generated by R* satisfies an n-Engel condition for some positive integer n if and only if R is m-Engel as a Lie ring for some positive integer m depending only on n.Received: 21 January 2003     

Semilocal rings with Engel conditions

The relation between the Engel structure of a semilocal ring and that of its multiplicative group is investigated. Suppose that every local ring whose multiplicative group satisfies an m-Engel condition for some positive integer m is an f (m)-Engel ring for some function f . It is proved that under this condition a corresponding statement holds for every semilocal ring which is generated by its multiplicative group. Received: 20 September 2005     

Minimal But Inefficient Presentations for Self Semidirect Products of the Free Abelian Monoid on Two Generators

Let R be a ring and M(R) the set consisting of zero and primitive idempotents of R. We study the rings R for which M(R) is multiplicative. It is proved that if R has a complete finite set of primitive orthogonal idempotents, then R is a finite direct product of connected rings precisely when M(R) is multiplicative. We prove that if R is a (von Neumann) regular ring with M(R) multiplicative, then every primitive idempotent in R is central. It is also shown that this does not happen even in semihereditary and semiregular rings. Let R be an arbitrary ring with M(R) multiplicative and e ∈ R be a primitive idempotent, then for every unit u ∈ R, it is proved that eue is a unit in eRe. We also prove that if M(R) is multiplicative, then two primitive idempotents e and f in R are conjugates, i.e., f = ueu ^?1 for some u ∈ U(R), if and only if ef ≠ 0.     

Extending the Concept of Chain Geometry

We introduce the chain geometry (K,R) over a ring R with a distinguished subfield K, thus extending the usual concept where R has to be an algebra over K. A chain is uniquely determined by three of its points, if, and only if, the multiplicative group of K is normal in the group of units of R. This condition is not equivalent to R being a K-algebra. The chains through a fixed point fall into compatibility classes which allow to describe the residue at a point in terms of a family of affine spaces with a common set of points.     

Topological nearrings whose additive groups are Euclidean

Let (G,+) be a group with a locally compact Hausdorff topology for which the binary operation + is continuous. Those, binary operation * onG for which (G, +, *) is a topological nearring are described. In the case whereG is abelian, those binary operations * for which (G, +, *) is a topological ring are also described. Versions of these results are then obtained in the special case where the group is the topological Euclideann-group,R ⁿ. A family of binary operations * for which (R ⁿ, +, *)_is a topological nearring is then investigated in some detail. Most of these nearrings turn out to be planar. Their ideals are completely determined and we characterize those nearrings which are simple. The multiplicative semi-groups (R ⁿ, *) of these nearrings are then investigated. Green's relations are completely determined and it is shown that a number of familiar properties of semigroups are equivalent for these particular semigroups. Finally, all those binary operations * for which (R, +, *) is a topological nearring are completely described. It is determined when any two of these nearrings are isomorphic and for each of these nearrings, its automorphism group, is completely determined.     

Commutative Semilocal Rings Whose Additive or Multiplicative Groups Have Finite Rank

ABSTRACT

Let R be an infinite semilocal ring. Then R is not finitely generated, neither the additive group R(+), nor the multiplicative group R? (of invertible elements) is minimax and at least one of these groups has infinite Prüfer rank.     

Artinian rings with nilpotent adjoint group

Let R be an Artinian ring (not necessarily with unit element), let Z(R) be its center, and let R° be the group of invertible elements of the ring R with respect to the operation a ∘ b = a + b + ab. We prove that the adjoint group R° is nilpotent and the set Z(R) + R° generates R as a ring if and only if R is the direct sum of finitely many ideals each of which is either a nilpotent ring or a local ring with nilpotent multiplicative group. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 417–426, March, 2006.     

Artinian rings with supersolvable adjoint group

The set of all elements of an associative ring R, not necessarily with a unit element, forms a monoid under the circle operation a ° b = a + b + ab. The group of all invertible elements of this monoid is called the adjoint group of R and is denoted by R ^°. It is proved that an artinian ring R with supersolvable adjoint group R ^° must be Lie supersolvable. An example of a Lie supersolvable ring with non-supersolvable adjoint group is also constructed. Received: 7 December 2007     

Loewy series,V-modules and trace ideals

If _R Vis a V-module and (R V W S) is a Morita context in which (S/WV) _s is flat, then the trace ideal WVis left V-module over S. If, in addition_S:(S/WV) is flat and S/WVis a fully left idempotent ring, then Sis also fully left idempotent. The lower (upper) Loewy length of _R Vprovides an upper bound for the corresponding Loewy length of _s(WV).     

Ring structures on groups of arithmetic functions

Using the theory of Witt vectors, we define ring structures on several well-known groups of arithmetic functions, which in another guise are formal Dirichlet series. The set of multiplicative arithmetic functions over a commutative ring R is shown to have a unique functorial ring structure for which the operation of addition is Dirichlet convolution and the operation of multiplication restricted to the completely multiplicative functions coincides with point-wise multiplication. The group of additive arithmetic functions over R also has a functorial ring structure. In analogy with the ghost homomorphism of Witt vectors, there is a functorial ring homomorphism from the ring of multiplicative functions to the ring of additive functions that is an isomorphism if R is a Q-algebra. The group of rational arithmetic functions, that is, the group generated by the completely multiplicative functions, forms a subring of the ring of multiplicative functions. The latter ring has the structure of a Bin(R)-algebra, where Bin(R) is the universal binomial ring equipped with a ring homomorphism to R. We use this algebra structure to study the order of a rational arithmetic function, as well the powersfα for α∈Bin(R) of a multiplicative arithmetic function f. For example, we prove new results about the powers of a given multiplicative arithmetic function that are rational. Finally, we apply our theory to the study of the zeta function of a scheme of finite type over Z.     

ON EXCHANGE RINGS WITH BOUNDED INDEX OF NILPOTENCE

This paper studies exchange rings R such that R/J(R) has bounded index of nilpotence. We give several characterizations of such rings. We prove that if a semiprimitive exchange ring R has index n, then for any maximal two-sided I of R, if R/I has length n, then there exists a central idempotent element e in R such that eRe is an n by n full matrix ring over some exchange ring with central idempotents, and the restriction π from eRe to R/I is surjective.     

On Finitely Annihilated Modules and Artinian Rings

Let R be a ring. An R-module M is finitely annihilated if the annihilator of M is the annihilator of a finite subset of M. It is proved that if R has right socle S then the ring R/S is right Artinian if and only if every singular right R-module is finitely annihilated. Moreover, a right Noetherian ring R is right Artinian if and only if every uniform right R-module is finitely annihilated. In addition, a (right and left) Noetherian ring is (right and left) Artinian if and only if every injective right R-module is finitely annihilated. This paper will form part of the Ph.D. thesis at the University of Glasgow of the second author. He would like to thank the EPSRC for their financial support     

ON NEAR-RINGS WITH MATRIX UNITS

Abstract

In ring theory it is well known that a ring R with identity is isomorphic to a matrix ring if and only if R has a set of matrix units. In this paper, the above result is extended to matrix near-rings and it is proved that a near-ring R with identity is isomorphic to a matrix near-ring if and only if R has a set of matrix units and satisfies two other conditions. As a consequence of this result several examples of matrix near-rings are given and for a finite group (Γ, +) with o(Γ) > 2 it is proved that M₀ (Γⁿ) is (isomorphic to) a matrix near-ring.     

On the Multiplicative Monoid of n×n Matrices Over Artinian Chain Rings

Let R be an Artinian chain ring with a principal maximal ideal. We investigate properties of matrices over R and give matrix representations of R-submodules of R ⁿ first, then consider Green's relations, Green's relation equivalent classes, Schützenberger groups of 𝒟-classes, principal factors, and group ?-classes of the multiplicative monoid M _n (R) of n × n matrices over R. Furthermore, we show that M _n (R) is an eventually regular semigroup and derive basic numerical information of M _n (R) when R is finite.     

PRIME IDEALS IN RINGS WITH INVOLUTION

Abstract

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.     

Global Theory of Lattice-Finite Noetherian Rings

We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal. Presented by H. Tachikawa     

Differential rings and ore extensions: Brown-McCoy rings

We consider here a ringK, a derivationD ofK and the differential polynomial ringR=K[X;D]. The ringK is said to be a Brown-McCoy ring if the prime radical coincides with the Brown-McCoy radical in every homomorphic image ofK. AD-Brown-McCoy ring is defined in a similar way. We prove the following conditions are equivalent: (i)K is aD-Brown-McCoy ring; (ii)R is a Brown-McCoy ring and for every maximal idealM ofR,K/(MνK) is aD-simple ring with 1. In addition, we give some applications and examples on the study of the transfer of the property of being a Brown-McCoy ring betweenK andR. Further, we study the relation between the prime and theD-prime ideals of a differential intermediate extension of a liberal extension. This paper was supported by a fellowship awarded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.     

Lifting Central Idempotents Modulo Jacobson Radicals and Ranks of K 0 Groups

Let R be a ring and let J(R) be the Jacobson radical of R. We discuss the problem of determining when the central idempotents in R/J(R) can be lifted to R. If R is a noetherian (artinian) ring, we give some conditions relative to the ranks of K ₀ groups under which the central idempotents in R/J(R) can be lifted. In particular, when R is semilocal, these conditions are necessary and sufficient. Moreover, we consider ranks of K ₀ groups of pullbacks of rings and obtain the upper and lower bounds on them under some suitable conditions.     

S-Noetherian Properties of Composite Ring Extensions

Let R be a commutative ring with identity and S a multiplicative subset of R. We say that R is an S-Noetherian ring if for each ideal I of R, there exist an s ∈ S and a finitely generated ideal J of R such that sI ? J ? I. In this article, we study transfers of S-Noetherian property to the composite semigroup ring and the composite generalized power series ring.