Full Matrix Ring Research Articles

The nil Temperley–Lieb algebra of type affine C

We introduce a type affine C analogue of the nil Temperley–Lieb algebra, in terms of generators and relations. We show that this algebra T(n), which is a quotient of the positive part of a Kac–Moody algebra of type Dn+1(2), has an easily described faithful representation as an algebra of creation and annihilation operators on particle configurations, reminiscent of the open TASEP model in statistical physics. The centre of T(n) consists of polynomials in a certain element Q, and T(n) is a free module of finite rank over its centre. We show how to localize T(n) by adjoining an inverse of Q, and prove that the resulting algebra is a full matrix ring over a ring of Laurent polynomials over a field. Although T(n) has wild representation type, over an algebraically closed field we can classify all the finite dimensional indecomposable representations of T(n) in which Q acts invertibly.

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.

Differential Inverse Power Series Rings with Quasi-Armendariz-like Condition

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, [Formula: see text]-quasi-Armendariz), is introduced and studied. It is shown that the [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime [Formula: see text]-quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given.

Quasi-normality of idempotents on nilpotents

We study the structure of idempotents in non-Abelian rings, concerning a ring property near to the normality of idempotents on the set of nilpotents. We call a ring with such property right idempotent-quasi-normalizing on nilpotents (simply, right IQNN), and study the structure of right IQNN rings in relation with matrix rings, polynomial ring, and factor rings, by which we extend the class of right IQNN rings. It is proved that the class of IQNN rings contains the 2 by 2 full matrix rings over fields and the upper triangular matrix rings over reduced rings. It is shown that given any countable field K, there exists a semiprime IQNN algebra R over K such that the polynomial ring R[x] over R is IQNN but not NI, and the upper nilradical of R[x] is zero.

π-Baer rings

We say a ring [Formula: see text] is [Formula: see text]-Baer if the right annihilator of every projection invariant left ideal of [Formula: see text] is generated by an idempotent element of [Formula: see text]. In this paper, we study connections between the [Formula: see text]-Baer condition and related conditions such as the Baer, quasi-Baer and [Formula: see text]-extending conditions. The [Formula: see text]-by-[Formula: see text] generalized triangular and the [Formula: see text]-by-[Formula: see text] triangular [Formula: see text]-Baer matrix rings are characterized. Also, we prove that a [Formula: see text]-by-[Formula: see text] full matrix ring over a [Formula: see text]-Baer ring is a [Formula: see text]-Baer ring. In contrast to the Baer condition, it is shown that the [Formula: see text]-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.

The number of ideals of [formula omitted] containing x(x − α)(x − β) with given index

It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let B denote an integral square matrix and 〈B〉 denote the subring of the full matrix ring generated by B. Then 〈B〉 is a free Z-module of finite rank, which guarantees that there are only finitely many ideals of 〈B〉 with given finite index. Thus, the formal Dirichlet series ζ〈B〉(s)=∑n≥1ann−s is well-defined where an is the number of ideals of 〈B〉 with index n. In this article we aim to find an explicit form of ζ〈B〉(s) when B has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, 〈B〉 is isomorphic to Z[x]/m(x)Z[x] where m(x) is the minimal polynomial of B over Q, and Z[x]/m(x)Z[x] is isomorphic to Z[x]/m(x+γ)Z[x] for each γ∈Z. Thus, the problem is reduced to counting the number of ideals of Z[x]/x(x−α)(x−β)Z[x] with given finite index where 0,α and β are distinct integers.

Modules with epimorphisms on chains of submodules

We say that an [Formula: see text]-module [Formula: see text] satisfies epi-ACC on submodules if in every ascending chain of submodules of [Formula: see text], except probably a finite number, each module in chain is a homomorphic image of the next one. Noetherian modules, semisimple modules and Prüfer [Formula: see text]-groups have this property. Direct sums of modules with epi-ACC on submodules need not have this property. If [Formula: see text] satisfies epi-ACC on submodules, then [Formula: see text] is quasi-Frobenius. As a consequence, a ring [Formula: see text] in which all modules satisfy epi-ACC on submodules is an artinian principal ideal ring. Dually, we say that an [Formula: see text]-module [Formula: see text] satisfies epi-DCC on submodules if in every descending chain of submodules of [Formula: see text], except probably a finite number, each module in chain is a homomorphic image of the preceding. Artinian modules, semisimple modules and free modules over commutative principal ideal domains are examples of such modules. A semiprime right Goldie ring satisfies epi-DCC on right ideals if and only if it is a finite product of full matrix rings over principal right ideal domains. A ring [Formula: see text] for which all modules satisfy epi-DCC on submodules must be an artinian principal ideal ring.

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.

The behavior of ascending chain conditions on submodules of bounded finite generation in direct sums

We construct a ring R which has the ascending chain condition on n-generated right ideals for each n ≥ 1 (also called the right pan-acc property), such that no full matrix ring over R has the ascending chain condition on cyclic right ideals. Thus, the right pan-acc property is not a Morita invariant. Moreover, a direct sum of (free) modules with pan-acc does not necessarily even have 1-acc.

Automorphisms of the zero-divisor graph of the full matrix ring

The zero-divisor graph of a non-commutative ring R, written as , is a directed graph with vertex set of all non-zero zero-divisors of R, and there is a directed edge from a vertex x to a distinct vertex y if and only if . Let M(n, q) (resp., T(n, q)) be the ring of all matrices (resp., upper triangular matrices) over a finite field . Recently, Wang (Linear Algebra Appl. 2015;465:214–220) determined the automorphisms of the zero-divisor graph of T(n, q). In this paper, we determine the automorphisms of , extending the result due to L. Wang from T(n, q) to M(n, q). Since the case is trivial, and the case has been examined in Ma et al. (J. Korean Math. Soc. 2016;53:519–532), we just determine the automorphisms of in the case. We show that a bijective map on is an automorphism of if and only if there exist invertible matrices and a such that for any , where , and depend on A, and .

ON A GENERALIZATION OF RIGHT DUO RINGS

We study the structure of rings whose principal right ideals contain a sort of two-sided ideals, introducing right �-duo as a generaliza- tion of (weakly) right duo rings. Abelian �-regular rings are �-duo, which is compared with the fact that Abelian regular rings are duo. For a right �-duo ring R, it is shown that every prime ideal of R is maximal if and only if R is a (strongly) �-regular ring with J(R) = N∗(R). This result may be helpful to develop several well-known results related to pm rings (i.e., rings whose prime ideals are maximal). We also extend the right �-duo property to several kinds of ring which have roles in ring theory. Throughout this note every ring is associative with identity unless otherwise specified. Given a ring R (possibly without identity), J(R), N∗(R), N ∗ (R), and N(R) denote the Jacobson radical, the prime radical, the upper nilradical (i.e., sum of all nil ideals), and the set of all nilpotent elements in R, respectively. It is well-known that N ∗ (R) ⊆ J(R) and N∗(R) ⊆ N ∗ (R) ⊆ N(R). We use R(x) (R((x))) to denote the polynomial (power series) ring with an indeterminate x over R. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). Use eij for the matrix unit with (i,j)-entry 1 and else- where 0. Denote {(aij) ∈ Un(R) | the diagonal entries of (aij) are all equal} by Dn(R). rR(−) (resp., lR(−)) is used to denote a right (resp., left) annihi- lator in R. Q denotes the direct product of rings. Z (Zn) denotes the ring of integers (modulo n). 1. Right �-duo rings In this section we introduce the concept of a right π-duo ring as a general- ization of weakly right duo ring, and study the structure of right π-duo rings. Let R be a ring and M be a right R-module. Buhphang and Rege (4) called M semicommutative if mRa = 0 whenever ma = 0 for m ∈ M and a ∈ R. We first consider the condition (∗): If ma = 0 for m ∈ M and a ∈ R, then mRa n = 0 for some n ≥ 1,

Quasi-Armendariz generalized power series rings

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We initiate the study of the [Formula: see text]-quasi-Armendariz condition on [Formula: see text], a generalization of the standard quasi-Armendariz condition from polynomials to skew generalized power series. The class of quasi-Armendariz rings includes semiprime rings, Armendariz rings, right (left) p.q.-Baer rings and right (left) PP rings. The [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. The [Formula: see text] formal upper triangular matrix rings of this class are characterized. We conclude some characterizations for a skew generalized power series ring to be semiprime, quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP. Examples to illustrate and delimit the theory are provided.

Annihilator Conditions related to the quasi-Baer Condition

We call a ring R an EGE-ring if for each $I \leq R$, which is generated by a subset of right semicentral idempotents there exists an idempotent $e$ such that $r(I) = eR$. The class EGE includes quasi-Baer, semiperfect rings (hence all local rings) and rings with a complete set of orthogonal primitive idempotents (hence all Noetherian rings) and is closed under direct product, full and upper triangular matrix rings, polynomial extensions (including formal power series, Laurent polynomials, and Laurent series) and is Morita invariant. Also we call $R$ an AE-ring if for each $I \unlhd R$, there exists a subset $S \subseteq S_{r}(R)$ such that $r(I) = r(RSR)$. The class AE includes the principally quasi-Baer ring and is closed under direct products, full and upper triangular matrix rings and is Morita invariant. For a semiprime ring $R$, it is shown that $R$ is an EGE (resp., AE)-ring if and only if the closure of any union of clopen subsets of $Spec(R)$ is open (resp., $Spec(R)$ is an EZ-space).

Factor rings of skew polynomial rings over a finite field

We consider the factor ring $$S = k[x; \sigma]/\langle x^{n} \rangle$$ , where k is a finite field. We show that this local ring can be represented by a matrix ring and any module over S is a direct sum of cyclic submodules. The ring S appears in Problem 2.45 of Dniester Notebook. For two such rings S 1 and S 2 we deal with left-right (S 1, S 2) bimodules. A certain factor ring of a skew polynomial ring over a Galois ring is also considered. This factor ring is isomorphic to a full matrix ring over some Galois ring.

When is a Sum of Annihilator Ideals an Annihilator Ideal?

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.

A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph - which we name as a truncated tree associated with A - whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant {\kappa}(A) for A and count the number of isomorphism classes of Leavitt path algebras with {\kappa}(A)=n. Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices.

Local derivations of full matrix rings

Let $${\mathcal{R}}$$ be a unital commutative ring and $${\mathcal{M}}$$ be a 2-torsion free central $${\mathcal{R}}$$ -bimodule. In this paper, for $${n \geqq 3}$$ , we show that every local derivation from M n ( $${\mathcal{R}}$$ ) into M n ( $${\mathcal{M}}$$ ) is a derivation.

STRUCTURE OF IDEMPOTENTS IN RINGS WITHOUT IDENTITY

We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring proper- ties pass to power series rings and polynomial rings. It is also shown that �-regular rings are strongly �-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommuta- tive rings appropriate for the situations occurred naturally in studying standard ring theoretic properties. 1. Definitions and notations Throughout this paper, R denotes an associative ring without identity, unless otherwise stated. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). We let eij denote the usual matrix units with 1 in the (i,j)-position and zeros elsewhere, if the base ring has identity 1. Denote {(aij) ∈ Un(R) | the diagonal entries of (aij) are all equal} by Dn(R). Zn denotes the ring of integers modulo n. GF(p n ) denotes the Galois field of order p n . J(R) denotes the Jacobson radical of R. | | denotes the cardinality. The characteristic of R is denoted by charR, and |a| denotes the order of a ∈ R in the additive subgroup of R generated by a. R + means the additive Abelian group (R,+). The polynomial ring with an indeterminate x over R is denoted by R(x). While speaking about minimal ring in a certain class of rings, we refer to a ring with minimal order for rings in that class, due to Xue (21). The notation (S) stands for the two-sided ideal of R generated by ∅ 6 S ⊆ R, and we also write (a1,...,an) in place of (S) for simplicity when S = {a1,...,an}. A ring is called Abelian if every idempotent is central. The zero in a nil ring is the only idempotent and so every nil ring is Abelian. The class of Abelian

Modules Whose Endomorphism Rings are Division Rings

The well-known Schur's Lemma states that the endomorphism ring of a simple module is a division ring. But the converse is not true in general. In this paper we study modules whose endomorphism rings are division rings. We first reduce our consideration to the case of faithful modules with this property. Using the existence of such modules, we obtain results on a new notion which generalizes that of primitive rings. When R is a full or triangular matrix ring over a commutative ring, a structure theorem is proved for an R-module M such that End R (M) is a division ring. A number of examples are given to illustrate our results and to motivate further study on this topic.

The moduli of representations with Borel mold

The author constructs the moduli of representations whose images generate the subalgebra of upper triangular matrices (up to inner automorphisms) of the full matrix ring for any groups and any monoids.