On the Behrens radical of matrix rings and polynomial rings

It is shown that the Behrens radical of a polynomial ring, in either commuting or non-commuting indeterminates, has the form of “polynomials over an ideal”. Moreover, in the case of non-commuting indeterminates, for a given coefficient ring, the ideal does not depend on the cardinality of the set of indeterminates. However, in contrast to the Brown-McCoy radical, it can happen that the polynomial ring R[X] in an infinite set X of commuting indeterminates over a ring R is Behrens radical while the polynomial ring R〈X〉 in an infinite set Y of non-commuting indeterminates over R is not Behrens radical. This is connected with the fact that the matrix rings over Behrens radical rings need not be Behrens radical. The class of Behrens radical rings, which is closed under taking matrix rings, is described.     

On the radicals of structural matrix rings

The relationship between the radical of a ringR and a structural matrix ring overR has been determined for some radicals. We continue these investigations, amongst others, determining exactly which radicals have the property (M(,R))=M( _s ,(R))+M( _a ,⁺(R))for any structural matrix ringM(,R) and finding (M(,R)) for any hereditary subidempotent radical .     

Strongly clean matrix rings over commutative local rings

We will completely characterize the commutative local rings for which M_n(R) is strongly clean, in terms of factorization in R[t]. We also obtain similar elementwise results which show additionally that for any monic polynomial f∈R[t], the strong cleanness of the companion matrix of f is equivalent to the strong cleanness of all matrices with characteristic polynomial f.     

Jordan isomorphisms of upper triangular matrix rings

Let R be a 2-torsionfree ring with identity 1 and let T_n(R), n ? 2, be the ring of all upper triangular n × n matrices over R. We describe additive Jordan isomorphisms of T_n(R) onto an arbitrary ring and generalize several results on this line.     

Jordan homomorphisms of upper triangular matrix rings

In this paper we investigate Jordan homomorphisms of upper triangular matrix rings and give a sufficient condition under which they are necessarily homomorphisms or anti-homomorphisms.     

A simple proof of a theorem on quasi-Baer rings

We give a simple proof of a theorem, due to Birkenmeier, Kim and Park, which states that if $R[x, x^{-1}]$ or $R[[x, x^{-1}]]$ is a quasi-Baer ring then R is a quasi-Baer ring. Received: 8 April 2002     

Morphic rings and unit regular rings

A ring R is called left morphic if for every a∈R. A left and right morphic ring is called a morphic ring. If M_n(R) is morphic for all n≥1 then R is called a strongly morphic ring. A well-known result of Erlich says that a ring R is unit regular iff it is both (von Neumann) regular and left morphic. A new connection between morphic rings and unit regular rings is proved here: a ring R is unit regular iff R[x]/(xⁿ) is strongly morphic for all n≥1 iff R[x]/(x²) is morphic. Various new families of left morphic or strongly morphic rings are constructed as extensions of unit regular rings and of principal ideal domains. This places some known examples in a broader context and answers some existing questions.     

Automorphism groups of generalized triangular matrix rings

We call a ring strongly indecomposable if it cannot be represented as a non-trivial (i.e. M≠0) generalized triangular matrix ring , for some rings R and S and some R-S-bimodule _RM_S. Examples of such rings include rings with only the trivial idempotents 0 and 1, as well as endomorphism rings of vector spaces, or more generally, semiprime indecomposable rings. We show that if R and S are strongly indecomposable rings, then the triangulation of the non-trivial generalized triangular matrix ring is unique up to isomorphism; to be more precise, if is an isomorphism, then there are isomorphisms ρ:R→R^′ and ψ:S→S^′ such that χ:=φ_∣M:M→M^′ is an R-S-bimodule isomorphism relative to ρ and ψ. In particular, this result describes the automorphism groups of such upper triangular matrix rings      

Some families of strongly clean rings

A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute with each other. For a commutative local ring R and for an arbitrary integer n?2, the paper deals with the question whether the strongly clean property of M_n(R[[x]]), , and M_n(RC₂) follows from the strongly clean property of M_n(R). This is ‘Yes’ if n=2 by a known result.     

Homomorphisms of matrix semigroups over division rings from dimension two to four

Let D be an arbitrary division ring and M_n(D) the multiplicative semigroup of all n×n matrices over D. We study non-degenerate, injective homomorphisms from M₂(D) to M₄(D). In particular, we present a structural result for the case when D is the ring of quaternions.     

Clean endomorphism rings

A ring is called clean if every element is the sum of an idempotent and a unit. It is shown that the endomorphism ring of a projective right module over a right perfect ring is clean.Received: 6 January 2003     

On the torsion theories of Morita equivalent rings

We generalize the well-known fact that for a pair of Morita equivalent ringsR andS their maximal rings of quotients are again Morita equivalent: If _n (M) denotes the torsion theory cogenerated by the direct sum of the firstn+1 injective modules forming part of the minimal injective resolution ofM then _n (R)= _n (S) where is the category equivalenceR-ModS-Mod. Consequently the localized ringsR _{n (R)} andS n _(S) are Morita equivalent.     

On functional identities and d-free subsets of rings. II

We show that a map in several variables on a prime ring satisfying an identity of polynomial type must be a quasi-polynomial (i.e., a polynomial in noncommutative variables whose coefficients are Martindale centroid valued functions)provided that the ring does not satisfy a standard identity of low degree. Obtained results have applications to the study of Lie maps of prime rings (Lie ideals of prime rings and skew elements of prime rings with involution)and to the study of Lie-admissible algebras and Lie homomorphisms of Lie algebras of Poisson algebras.     

Nil-clean matrix rings

We characterize the nil-clean matrix rings over fields. As a by product, we obtain a complete characterization of the finite rank Abelian groups with nil-clean endomorphism ring and the Abelian groups with strongly nil-clean endomorphism ring, respectively.     

Subalgebras of matrix algebras generated by companion matrices

Let f,g∈Z[X] be monic polynomials of degree n and let C,D∈M_n(Z) be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra Z〈C,D〉 to be a sublattice of finite index in the full integral lattice M_n(Z), in which case we compute the exact value of this index in terms of the resultant of f and g. If R is a commutative ring with identity we determine when R〈C,D〉=M_n(R), in which case a presentation for M_n(R) in terms of C and D is given.     

Lie properties of symmetric elements in group rings II

Let F be a field of characteristic different from 2, and G a group with involution ∗. Write (FG)⁺ for the set of elements in the group ring FG that are symmetric with respect to the induced involution. Recently, Giambruno, Polcino Milies and Sehgal showed that if G has no 2-elements, and (FG)⁺ is Lie nilpotent (resp. Lie n-Engel), then FG is Lie nilpotent (resp. Lie m-Engel, for some m). Here, we classify the groups containing 2-elements such that (FG)⁺ is Lie nilpotent or Lie n-Engel.     

On injective Jordan semi-triple maps of matrix algebras

We show that every injective Jordan semi-triple map on the algebra M_n(F) of all n × n matrices with entries in a field F (i.e. a map Φ:M_n(F)→M_n(F) satisfying

Φ(ABA)=Φ(A)Φ(B)Φ(A)