Full Matrix Ring Research Articles

Group Identities and Prime Rings Generated by Units

Let R be a prime ring and U(R) the group of units of R. We prove that if U(R) generates R and satisfies a group identity,then R is either a domain or a full matrix ring over a finite field.

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.

Group Gradings on Full Matrix Rings

We study G-gradings of the matrix ring Mn(k), k a field, and give a complete description of the gradings where all the elements ei,j are homogeneous, called good gradings. Among these, we determine the ones that are strong gradings or crossed products. If G is a finite cyclic group and k contains a primitive |G|th root of 1, we show how all G-gradings of Mn(k) can be produced. In particular we give a precise description of all C2-gradings of M2(k) and show that for algebraically closed k, any such grading is isomorphic to one of the two good gradings.

Whitehead test modules

A (right R-) module N is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module M , ExtR(M,N) = 0 implies M is projective. Dually, i-test modules are defined. For example, Z is a p-test abelian group iff each Whitehead group is free. Our first main result says that if R is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH. On the other hand, for any ring R, there is a proper class of i-test modules. Dually, there is a proper class of p-test modules over any right perfect ring. A non-semisimple ring R is said to be fully saturated (κ-saturated) provided that all non-projective (≤ κ-generated non-projective) modules are i-test. We show that classification of saturated rings can be reduced to the indecomposable ones. Indecomposable 1-saturated rings fall into two classes: type I, where all simple modules are isomorphic, and type II, the others. Our second main result gives a complete characterization of rings of type II as certain generalized upper triangular matrix rings, GT (1, n, p, S, T ). The four parameters involved here are skew-fields S and T , and natural numbers n, p. For rings of type I, we have several partial results: e.g. using a generalization of Bongartz Lemma, we show that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. In several recent papers, our results have been applied to Tilting Theory and to the Theory of ∗-modules. In modern algebra, the structure of rings, R, is studied by means of properties of corresponding module categories, Mod-R. In most cases, it is not possible to characterize Mod-R fully. Nevertheless, there are important subclasses of Mod-R that can be treated in detail and that shed light on the whole of Mod-R. Among the prominent ones are the classes of all projective and all injective modules. Recall that a module M is said to be projective (injective) provided that the functorHomR(M,−) (HomR(−,M)) preserves short exact sequences. There is also a universal algebraic aspect: each module is a factor module of a projective module, and a submodule of an injective module. So a possible strategy to investigate ModR consists in describing all injective modules, and for each injective module, I, all its submodules. The first step is usually relatively easy, but the second may be quite hard. For example, using this strategy for abelian groups, one meets serious difficulties already for I = Q⊕Q (see e.g. [E, Theorem 2]). Received by the editors March 17, 1995. 1991 Mathematics Subject Classification. Primary 16E30; Secondary 03E35, 20K35.

A criterion for isomorphic tiled orders over a local Dedekind domain

Throughout this note R will denote a local Dedekind domain with a unique maximal ideal zR and the quotient ring K. Let n(^2) be an integer. As usual, an i?-order A in the full nXn matrix ring (K)n over K is an i?-subalgebra of (K)n such that A is finitely generated as an i?-module and AK=(K)n. An i?-orderin (K)n is called tiledif it contains n orthogonal idempotents. If A is a tiled Z?-order in (K)n then up to conjugation we may assume that it contains the idempotents en (l^z^n), where {etj\l which is constructed from their graph theoretic properties. In this note we shall remove the hypothesis between (xR)n and (/?), and prove the following

Balanced modules and rings of left colocal type

is a ring homomorphism, where rm is the left multiplication of r, i.e., rm(x)= rx for all x^M. If p is surjective, or equivalently every element of BiEndG?M) is a left multiplication of an element of R, then M is said to be balanced. If every finitely generated faithfulleft i?-module is balanced, then R is called a left QF-l ring. Further, R is said to be a left balanced ring if R/l is left QF-l for every two sided ideal / of R; this condition is equivalent to the condition that every finitelygenerated left module is balanced. The concept of QF-l rings was introduced by Thrall [18] as a generalization of quasi-Frobenius rings, and in the same paper he proposed to give an internal characterization of QF-l rings. At the present time, however, this problem is not solved completely, though partial answers are given in several cases (Camillo and Fuller [1], Dlab and Ringel [2], Fuller [7], Makino [8, 9, 10], Morita [11], Ringel [12] and Tachikawa [17]). On the other hand, the structure of balanced rings was completely determined by Dlab and Ringel [2, 3, 4, 5, 6]. Indeed they proved that an indecomposable ring is left balanced if and only if it is a full matrix ring over either a local uniserial ring or an exceptional ring. We should note that exceptional rings are characterized as local rings of either left or right colocal type with the square zero radical (see Section 1 for the precise definition of rings of left colocal type). Our aim of this paper is to show that in the case of left serial rings, left QF-lness, which is much more weaker condition than left balancedness, implies rings of left colocal tvoe (Theorem). For the case of a finite dimensional

Products of idempotent integer matrices

AbstractLet E denote the set of non-identity idempotent matrices in the full matrix ring Mn(R) over a principal ideal domain R. A necessary and sufficient condition is found for the subsemigroup generated by E to be the set of all matrices in Mn(R) of rank less than n. The condition is satisfied when R is a discrete valuation ring and when R is the ring of integers. Thus every n × n matrix of rank less than n is a product of idempotent integer matrices.

Row rank and the structure of matrix subrings

Subrings of full matrix rings which have positive row or column rank can be approximated by rings of triangular matrices.

Polynomial automorphisms of lattices

It is well known that every automorphism of the symmetric group Sn, n # 6, is inner, that is, of the form x ~ a l xa for some given a ~ Sn. A similar result holds for full matrix rings over a commutative field. It is also known that each group G can be embedded in a group H in such a way that every automorphism of G is induced by an inner automorphism of H. At first sight this result cannot be transferred to lattices. Indeed, there are only four binary lattice terms on a and x: a, x, a v x and a ^ x, and if any of these yield an isomorphism it must be the identity. Consequently we must "generalize" the concept of inner automorphism for lattices. We thus propose the following, due to Dietmar Schweigert [5]:

Representation of tiled matrix rings as full matrix rings

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and setThen T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).

Locally nilpotent skew linear groups

Throughout this paper D denotes a division ring with centre F and n a positive integer. A subgroup G of GL(n,D) is absolutely irreducible if the F-subalgebra F[G] enerated by G is the full matrix ring Dn ×n. It is completely reducible (resp. irreducible) if row n-space Dn over D is completely reducible (resp. irreducible), as D–G bimodule in the obvious way. Absolutely irreducible skew linear groups have a more restricted structure than irreducible skew linear groups, see for example [7],[8], [8] and [10]. Here we make a start on elucidating the structure of locally nilpotent suchgroups.

Soluble normal subgroups of skew linear groups

A skew linear group is a subgroup of GL(n,D) for some positive integer n and division ring D. In [13] and [14] we studied a locally finite normal subgroup H of a skew linear group G and found in particular that the structure of G/Cc(H) is very restricted. Here we consider corresponding questions concerning soluble normal subgroups of G. The results are similar to those of [13] and [14], but the conclusions are slightly weaker, necessarily so as we show with examples. Let F be the centre of the division ring D. We say that the subgroup G of GL(n, D) is absolutely irreducible if the subring F[G] of the matrix ring D nxn generated by F and G is the full matrix ring. See [1 3] for the background concerning this concept. The locally finite results of [13] and the soluble results here can be combined, and it is in this form that we state our conclusions. Thus we consider the class P(gA U L~) of all groups with a series of finite length whose factors are abelian or locally finite. (In general we use Hall's calculus of group classes as expounded in [4, Chapter 1] except that we use Q in place of H.) Our main theorem is the following.

Modules over prime Krull rings

If one adds the assumptions that (i) R is a domain and (ii) K dim, = 1 then this result has been proved by Chamarie [Z] and our proof is a refinement of his. In Chamarie’s proof, condition (ii) arises from the fact that he proves the result by passing to a quotient category of the category of R-modules. This condition can be avoided by proving the result back in the category of R-modules. To generalise the result from domains to prime rings is technically more involved, but the idea is easy enough: given a reflexive prime ideal P of R, the local ring R, is a full matrix ring over a domain D and so one should try to make use of this domain D. This is done in Section 2 and used in Section 3 to generalise Bourbaki’s Theorem. Section 1 contains some elementary results about Krull rings. In fact, we prove a slight generalisation of Bourbaki’s Theorem, which is concerned with giving the appropriate short exact sequence for our module M and a submodule N. In particular, this has the following result as a consequence.

Auslander-reiten quivers of schurian orders

We shall consider orders ⋀ over a complete discrete rank one valuation ring R in a split full matrix ring containing a complete sex of primitive orthogonal idempot ents. In case s of finite lattice type and R is the power series ring in one variable over its residexe class field k , we give a description of its index composable lattices and its Auslander-Reiten quiver in terms of representations of partially ordered sets. By a model theoretic argument, this implies a description of all indecomposable lattices if R is the completion of the ring of integers in an algebraic number field at all but possibly a finite number of primes.

Description of subgroups of the full linear group over a semilocal ring that contain the group of diagonal matrices

It has been proved (Ref. Zh. Mat., 1978, 9A237) that for a semilocal ring Λ in which each residue field of the center contains at least seven elements we have the following description of subgroups of the full linear group GL(n,Λ) that contain the group of diagonal matrices: for each such subgroup H there is a uniquely definedD -net of idealsS (Ref. Zh. Mat., 1977, 2A288) such that G(S)≤H≤N(S), ,whereN(S) is the normalizer of theS -net subgroup G(S). It is noted that this result is also true under the following weaker assumption: a decomposition of a quotient ring of the ring Λ into a direct sum of full matrix rings over skew fields does not contain skew fields with centers of less than seven elements or the ring of second-order matrices over the field of two elements.

THE STRUCTURE OF SEMIPERFECT RINGS WITH COMMUTATIVE JACOBSON RADICAL

Let be a semiperfect ring with commutative Jacobson radical , and let , where the are the full matrix rings over skew fields . In this article we prove theorems which enable us to reduce the study of the structure of to the study of the structure of local commutative rings for which each is a field which is a finite Galois extension of its prime subfield. Bibliography: 7 titles.

A note on Noetherian orders in Artinian rings

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

The Schur Group of an Algebraic Number Field

If G denotes a finite group and Q the rational field, then the group algebra Q[G] is a direct sum of simple rings, each a full matrix ring over a division ring. One may ask which simple rings arise this way as G is allowed to range over all finite groups. The center of such a simple ring is a field K which is a subfield of Q(sm), where sm is a primitive mth root of unity and m divides the order of G. If we fix our attention on one particular field K, the subset of the Brauer group B(K) of K consisting of all those classes containing an algebra which is isomorphic to a simple summand of Q[G] for some G is a subgroup, S(K), the Schur group of K. This is the same group as the set of classes in B(K) which contain a simple summand of K[G] for some G. We take this as the definition of S(K) when K is any field. Since the elements of B(K) are uniquely determined by Hasse invariants when K is an algebraic number field, it is natural to ask for a description of S(K) in these terms. The main purpose of this paper is to give a description of S(K)p, the p-primary subgroup of S(K), in the case p is odd and K is any abelian extension of Q. In other words we describe up to equivalence in the Brauer group, those simple algebras of odd index which can appear as a direct summand of Q[G]. Some information about the 2-primary part of S(K) is also obtained in the case K contains s, a fourth root of unity. For such a field we determine which numbers can occur as a local index of an element in S(K). Let Q(sm) = L be the least root of unity field containing the given K and let R be the set of rational primes dividing m which ramify between K and L with ramification index divisible by the prime p. Let S(K, R)p denote the subgroup of S(K)p consisting of all elements split at all primes of K which do not divide an element of A. For a prime q not dividing m, let S(K, q)p denote the subgroup of S(K)p of elements split everywhere except at the divisors of q. We obtain the following result.

Rings whose faithful left ideals are cofaithful

A left module M over a ring R is cofaithful in case there is an embedding of R into a finite product of copies of M. Our main result states that a semiprime ring R is left Goldie, that is, has a semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left ideal. The proof is elementary and does not make use of the Goldie and LesieurCroisot theorems. We show that (i) and (ii) are Morita invariant. Moreover, (ii) is invariant under polynomial extensions, and so is (i) for commutative rings. Absolutely torsionfree rings are studied. The ring Q is a left classical quotient ring for the ring R C Q if every regular element (nondivisor of zero) of R is invertible in Q and if every element of Q is of the form b ~ιa where a,b E R and b is regular; in this case we also say that R is a left order in Q. A ring is said to be left Goldie if it has the ascending chain condition on left annihilators and has finite uniform dimension. (A left R -module has finite uniform dimension if it has no infinite direct sum of nonzero submodules, and it is said to be uniform if it is nonzero and any two nonzero submodules have a nontrivial intersection.) A theorem of Goldie [8,9] and Lesieur and Croisot [12] states that a ring is a left order in a semisimple Artinian ring if and only if it is semiprime and left Goldie. It is known that the ascending chain condition on left annihilators is not preserved under an equivalence of categories (Morita invariant); in fact, it does not go up to matrix rings. It is unknown whether being left Goldie is Morita invariant. In section two we give a proof of the theorem stated in the abstract, and in the prime case we give a proof which shows directly that such a ring is an order in a full matrix ring over a division ring. We also weaken the hypothesis of an important theorem on semiprime PI rings. In the third section we use these techniques to study absolutely torsion-free rings. In particular, we show that an absolutely torsionfree ring is Goldie if and only if it has a uniform left ideal, and that the endomorphism ring of a finitely generated projective module over an absolutely torsion-free ring is absolutely torsion-free.

One-Sided Inverses in Rings

Following Herstein [2], we will call a ring R with identity von Neumann finite (vNf) provided that xy = 1 implies yx — WnR. Kaplansky [4] showed that group algebras over fields of characteristic zero are vNf rings, and further, that full matrix rings over such rings are also vNf. Herstein [2] has posed the problem for group algebras over fields of arbitrary characteristic. If group algebras over fields are always vNf, then it is easily seen that group algebras over commutative rings are always vNf. What conditions on the underlying ring of scalars would force the vNf property for all group rings over it?