Structural Matrix Rings Research Articles

Ring isomorphic to a structural matrix ring over a division ring

Ring isomorphic to a structural matrix ring over a division ring

Induced good gradings of structural matrix rings

Our approach to structural matrix rings defines them over preordered directed graphs. A grading of a structural matrix ring is called a good grading if its standard unit matrices are homogeneous. For a group G, a G-grading set is a set of arrows with the property that any assignment of these arrows to elements of G uniquely determines an induced good grading. One of our main results is that a G-grading set exists for any transitive directed graph if G is a group of prime order. This extends a result of Kelarev. However, an example of Molli Jones shows there are directed graphs which do not have G-grading sets for any cyclic group G of even order greater than 2. Finally, we count the number of nonequivalent elementary gradings by a finite group of a full matrix ring over an arbitrary field.

Radicals of matrix rings generated by a companion matrix

ABSTRACTThe radicals of full matrix rings as well as structural matrix rings have been studied extensively and there is a well-developed theory for both. Here we initiate the radical theory for another class of matrix rings over a given ring; the matrix ring generated by the companion matrix of a polynomial over the ring.

Automorphisms of some structural infinite matrix rings

Automorphisms of some structural infinite matrix rings

Jordan isomorphisms of generalized structural matrix rings

We describe the sub-bimodules of matrix bimodules over two structural matrix rings. Structural matrix bimodules arise as particular such sub-bimodules, and we discuss when such a bimodule is faithful or indecomposable. As an application, we obtain a large class of rings whose Jordan isomorphisms are either ring isomorphisms or ring anti-isomorphisms. Complete upper block triangular matrix rings over 2-torsion-free indecomposable rings are elements of this class.

On p.p. structural matrix rings

A ring is called a left p.p. ring if every principal left ideal is projective. The objective here is to completely determine the left p.p. structural matrix rings over a von Neumann regular ring.

OPTIMIZATION OF MATRIX SEMIRINGS FOR CLASSIFICATION SYSTEMS

AbstractThe max-plus algebra is well known and has useful applications in the investigation of discrete event systems and affine equations. Structural matrix rings have been considered by many authors too. This article introduces more general structural matrix semirings, which include all matrix semirings over the max-plus algebra. We investigate properties of ideals in this construction motivated by applications to the design of centroid-based classification systems, or classifiers, as well as multiple classifiers combining several initial classifiers. The first main theorem of this paper shows that structural matrix semirings possess convenient visible generating sets for ideals. Our second main theorem uses two special sets to determine the weights of all ideals and describe all matrix ideals with the largest possible weight, which are optimal for the design of classification systems.

Invertibility and Dedekind finiteness in structural matrix rings

An example in Szigeti and van Wyk [J. Szigeti and L. van Wyk, Subrings which are closed with respect to taking the inverse, J. Algebra 318 (2007), pp. 1068–1076] suggests that Dedekind finiteness may play a crucial role in a characterization of the structural subrings M n (θ, R) of the full n × n matrix ring M n (R) over a ring R, which are closed with respect to taking inverses. It turns out that M n (θ, R) is closed with respect to taking inverses in M n (R) if all the equivalence classes with respect to θ ∩ θ−1, except possibly one, are of a size less than or equal to p (say) and M p (R) is Dedekind finite. Another purpose of this article is to show that M n (θ, R) is Dedekind finite if and only if M m (R) is Dedekind finite, where m is the maximum size of the equivalence classes (with respect to θ ∩ θ−1). This provides a positive result for the inheritance of Dedekind finiteness by a matrix ring (albeit not a full matrix ring) from a smaller (full) matrix ring.

Some remarks on structural matrix rings and matrices with ideal entries

Associating to each pre-order on the indices 1,...,n the corresponding structural matrix ring, or incidence algebra, embeds the lattice of n-element pre-orders into the lattice of n x n matrix rings. Rings within the order-convex hull of the embedding, i.e. matrix rings that contain the ring of diagonal matrices, can be viewed as incidence algebras of ideal-valued, generalized pre-order relations. Certain conjugates of the upper or lower triangular matrix rings correspond to the various linear orderings of the indices, and the incidence algebras of partial orderings arise as intersections of such conjugate matrix rings.

Left Self Distributively Generated Structural Matrix Rings#

ABSTRACT Let K denote a commutative ring with unity and A be a K-algebra. An element, d ∈ A is said to be left self distributive, or LSD, if dxy = dx dy for all x, y ∈ A. Let ℒ(A) be the set of LSD elements. Similarly, one can define the set of right self distributive, or RSD, elements and let ℛ(A) be the set of RSD elements. Let 𝒟(A) = ℒ(A) ∩ ℛ(A), the set of self distributive, or SD, elements. An algebra, A, is said to be left self distributively generated, or LSD-generated, if A = mod K (ℒ(A)), the K-module generated by ℒ(A). Analogously, one defines RSD-generated and SD-generated algebras. If A = mod K (ℒ(A)) = mod K (ℛ(A)), then A is said to be LSD/RSD-generated, which is a strictly larger class than the class of SD-generated algebras. Examples are given to illustrate the variety of LSD-generated algebras. This paper continues the study of LSD-generated, RSD-generated, LSD/RSD-generated and SD-generated algebras. This paper characterizes exactly which structural matrix rings are LSD-generated. The paper begins with an important lemma that characterizes LSD elements in a matrix ring in terms of the entries of the matrix. The main result characterizes those structural matrix rings that are LSD-generated, first in terms of a 2 × 2 generalized matrix ring, then strictly in terms of the shape of the matrix ring. Sharper results are obtained for LSD/RSD-generated and SD-generated structural matrix rings. The final section is devoted to an application of this result to endomorphism rings. If the endomorphism ring of a finitely generated module is a homomorphic image of a structural matrix ring, then the module is a direct sum of cyclic modules. Further conditions are given to describe when the structural matrix ring is LSD-generated, in terms of the annihilators of the generating set.

Link between a natural centralizer and the smallest essential ideal in structural matrix rings

In a structural matrix ring Mn (ρ R) over an arbitrary ring R we determine the centralizer of the set of matrix units in Mn (ρR) associated with the anti-symmetric part of the reflexive and transitive binary relation ρ on {1,2,…,n}. If the underlying ring R has no proper essential ideal, for example if R is a field, then we show that the largest ideal of Mn (ρR) contained in the mentioned centralizer coincides with the smallest essential ideal of Mn (ρR).

Generalized blocked triangular matrix rings associated with finite abelian centralizer near-rings

For N any member of a large class of finite abelian right centralizer near-rings, the subring of the ring End(N) of endomorphisms of (N, +) generated by the set of right multiplication maps on N is explicitly described as a generalized blocked triangular matrix ring, which in some cases turns out to be a structural matrix ring.

Matrix rings satisfying column sum conditions versus structural matrix rings

The internal characterization of a structural matrix ring in terms of a set of matrix units associated with a quasiorder relation is used to obtain isomorphisms between seemingly different classes of subrings of a complete matrix ring.

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 γ have the property γ(M(ρ,R))=M(ρs,γ(R))+M(ρa,γ+(R))for any structural matrix ringM(ρ,R) and finding β(M(ρ,R)) for any hereditary subidempotent radical β.

Do isomorphic structural matrix rings have isomorphic graphs?

We first provide an example of a ring R R such that all possible 2 × 2 2\times 2 structural matrix rings over R R are isomorphic. However, we prove that the underlying graphs of any two isomorphic structural matrix rings over a semiprime Noetherian ring are isomorphic, i.e. the underlying Boolean matrix B B of a structural matrix ring M ( B , R ) \mathbb M(B,R) over a semiprime Noetherian ring R R can be recovered, contrary to the fact that in general R R cannot be recovered.

MATRIX RINGS OF INVARIANT SUBSPACES

Abstract Consider a set V of R-submodules of R n, R a commutative ring. We provide a sufficient condition on V such that the ring of all R-endomorphisms of R n, which leave every R-submodule in V invariant, is a structural matrix ring. We also obtain results suggesting strongly that the mentioned condition is necessary too in case R is a field.

Polynomial regularities in structural matrix rings

It is known that many special radicals R satisfy the matrix condition an arbitrary ring, but that these radicals do not carry over from the base ring R to structural matrix rings in this natural way. In this paper we study the radicals (of structural matrix rings) determined by some polynomial regularities. We obtain various classes of polynomial regularities and classes of structural matrix rings for which the corresponding radicals behave differently from special radicals. For example, on the one hand, some radicals carry over to certain structural matrix rings in the mentioned natural way, whereas on the other hand it might happen that the radical of a structural matrix ring, which is not a complete matrix ring, is {0}, in spite of the fact that sucha radical satisfies the matrix condition, for example the Von Neumann radical. It turns out that the presence or absence of reflexivity and antisymmetry of the underlying Boolean matrix of a structural matrix ring plays a crucial role in this regard.

Aautomorphism groups of certain structural matrix rings

Aautomorphism groups of certain structural matrix rings

An internal characterisation of structural matrix rings

It is known that structural matrix rings pro-vide a natural passage from complete matrix rings to upper and lower triangular matrix rings, and they often explain the peculiarities regarding certain properties of complete matrix rings on the one hand and of triangular matrix rings on the other hand. In this paper the concept of a set of matrix units in a ring associated with a quasi-order relation is introduced and used to provide an internal char-acterisation of structural matrix rings.

RADICALS OF STRUCTURAL MATRIX RINGS

The concept of a structural matrix ring was introduced by van Wyk [8]. In [9] certain radicals of these matrix rings are studied. The purpose of this paper is to obtain similar results for a wider class of radicals.