Semisimple Rings Research Articles

Cyclotomic cosets and primitive idempotents in semisimple ring

Let be distinct primes. Let , where at least two of αi’ s are nonzero positive integers. In this article, it is shown that with the help of λ-mapping it is easy to count the number of q-cyclotomic cosets modulo m. The explicit expressions of primitive idempotents are obtained in the semisimple ring Rm over Fq taking q as a primitive root (or a quadratic residue) modulo pi under the condition that , where denote the order of q modulo pi.

Prime virtually semisimple modules and rings

This article is a sequel to the recent three papers on “virtually semisimple modules and rings,” by Behboodi et al., which two of them appeared in the Algebras and Representation Theory and Communications in Algebra in 2018. An R-module M is called virtually semisimple if each submodule of M is isomorphic to a direct summand. A ring R is called left (resp., right) virtually semisimple if (resp., RR) is virtually semisimple. In this article, we study rings and modules in which every prime submodule is isomorphic to a direct summand, and called them prime virtually (or -virtually) semisimple modules. A ring R is called left (resp., right) -virtually semisimple if (resp., RR) is -virtually semisimple. The results of the article are inspired by a characterization of left -virtually semisimple rings. We prove that these rings are precisely the left virtually semisimple rings, and in this case , where each Di is a domain and each is a principal left ideal ring. We also answer to the following questions: (i) Describe rings R where each (finitely generated or cyclic) left R-module is -virtually semisimple?, and (ii) Describe rings R where each left R-module is a direct sum of indecomposable -virtually semisimple modules? Finally, we study -virtually semisimple modules over commutative rings.

Rings with only finitely many essential right ideals

In this paper, we study rings with only finitely many essential right ideals (right fe-rings for short). We see that these rings have some similar properties of semisimple rings. In fact, we prove that R is a right fe-ring if and only if where each Ai is a right artinian right uniserial ring and B is a finite ring. Consequently, we show that a right fe-ring R is right artinian if and only if it is right noetherian. We prove that right duo right fe-rings are zero-dimensional. Some complete properties of prime right fe-rings are given. We prove that if R is a right fe-ring, then J(R) and Z(RR) are nilpotent ideals and is finite.

On the trace graph of matrices

Let R be a commutative ring with identity, $${M_n(R)}$$ be the set of all $${n \times n}$$ matrices over R and $${M_n(R) ^{*} }$$ be the set of all non-zero matrices of $${M_n(R)}$$ where $${n \geq 2}$$ . For a matrix $${A \in M_n(R)}$$ , $${{\rm Tr} (A)}$$ is the trace of A. The trace graph of the matrix ring $${M_n(R)}$$ , denoted by $${\Gamma_t(M_n(R))}$$ , is the simple undirected graph denoted by $${\Gamma_t(M_n(R))}$$ with vertex set $$\{{A \in M_n(R) ^{*} : }$$ there exists $${B \in M_n(R) ^{*} }$$ such that $${{\rm Tr}(AB)=0}\}$$ and two distinct vertices A and B are adjacent if and only if $${{\rm Tr} (AB) = 0}$$ . First, we prove that $${\Gamma_t(M_n(R))}$$ is 2-connected and hence obtain Eulerian properties of $${\Gamma_t(M_n(R))}$$ . Also we obtain the domination number of $${\Gamma_t(M_n(R))}$$ of a commutative semisimple ring R and obtain the domination number for $${\Gamma_t(M_n(\mathbb Z_2^m))}$$ . Finally, it is proved that for a commutative ring R with identity, $${\Gamma_t(M_n(R))}$$ is non-planar and classified all finite commutative rings R with identity for which the trace graph has thickness 2.

Some cyclic codes of length ${8p^n}$ over $GF(q)$, where order of $q$ modulo $8p^n$ is $\frac{\phi{(p^n)}}{2}$

Let $G$ be a finite group and $F$ be finite field of prime power order $q$ (of type $8k+5$) and order of $q$ modulo $8p^n$ is $\frac{\phi(p^n)}{2}$. If $p$ is prime of type $4k+1$, then the semi-simple ring $R_{8p^n}\equiv \frac{GF(q)[x]}{}$ has $16n+6$ primitive idempotents and for $p$ of type $4k+3$, then $R_{8p^n}$ has $12n+6$ primitive idempotents. The explicit expression for these idempotents are obtained, the generating polynomials and minimum distance bounds for cyclic codes are also completely described.

2-Nilpotent-invariant modules

A module which is invariant under automorphisms of its injective envelope is called an automorphism-invariant module. The class of automorphism-invariant modules was introduced and investigated by Lee and Zhou in 2013. In this paper, we study the class of modules which are invariant under all nilpotent endomorphisms of their injective envelopes of index two, such as modules are called 2-nilpotent-invariant. Many basic properties are obtained. For instance, it is proved that a nonsingular module [Formula: see text] is a weak duo 2-nilpotent-invariant module if and only if [Formula: see text] is a strongly regular ring. For the ring [Formula: see text] satisfying every cyclic right [Formula: see text]-module is 2-nilpotent-invariant, we prove that [Formula: see text], where [Formula: see text] are rings which satisfy [Formula: see text] is a semi-simple Artinian ring and [Formula: see text] is square-free as a right [Formula: see text]-module and all idempotents of [Formula: see text] is central.

Small potential counterexamples to the pure semisimplicity conjecture

The pure semisimplicity conjecture or pssc states that every left pure semisimple ring has finite representation type. Let [Formula: see text] be division rings, and assume we identify conditions on a [Formula: see text]-[Formula: see text]-bimodule [Formula: see text] which are sufficient to make the triangular matrix ring [Formula: see text] into a left pure semisimple ring which is not of finite representation type. It is then said that those conditions yield a potential counterexample to the pssc. Simson [17–20] gave several such conditions in terms of the sequence of the left dimensions of the left dual bimodules of [Formula: see text]. In this paper, conditions with the same purpose are given in terms of the continued fraction attached to [Formula: see text], and also through arithmetical properties of a division ring extension [Formula: see text].

On Boolean Lattices of Module Classes

Some properties of and relations between several (big) lattices of module classes are used in this paper to obtain information about the ring over which modules are taken. The authors reach characterizations of trivial rings, semisimple rings and certain rings over which every torsion theory is hereditary.

Maximal Subrings and Covering Numbers of Finite Semisimple Rings

ABSTRACTWe classify the maximal subrings of the ring of n×n matrices over a finite field, and show that these subrings may be divided into three types. We also describe all of the maximal subrings of a finite semisimple ring, and categorize them into two classes. As an application of these results, we calculate the covering number of a finite semisimple ring.

On semi-projective modules and their endomorphism rings

This paper provides the several homological characterization of perfect rings and semi-simple rings in terms of semi-projective modules. We investigate whether Hopkins–Levitzki Theorem extend to semi-projective module i.e. whether there exists an artinian semi-projective module which are noetherian. Unfortunately, the answer we have is negative; counter example is given. However, it is shown that the answer is positive for certain large classes of semi-projective modules in Proposition 2.26. We have discussed the summand intersection property, summand sum property for semi-projective modules. Apart from this, we have introduced the idea of [Formula: see text]-hollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi-projective modules is a local ring.

Leavitt path algebras: Graded direct-finiteness and graded Σ-injective simple modules

In this paper, we give a complete characterization of Leavitt path algebras which are graded Σ-V rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra L over an arbitrary graph E is a graded Σ-V ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over K or K[x,x−1] with appropriate matrix gradings. We also obtain a graphical characterization of such a graded Σ-V ring L. When the graph E is finite, we show that L is a graded Σ-V ring ⟺L is graded directly-finite ⟺L has bounded index of nilpotence ⟺ L is graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph E is infinite. Following this, we also characterize Leavitt path algebras L which are non-graded Σ-V rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra L is a graded Σ-V ring, then L is always graded directly-finite. Examples show the subtle differences between graded and non-graded directly-finite rings. Leavitt path algebras which are graded directly-finite are shown to be directed unions of graded semisimple rings. Using this, we give an alternative proof of a theorem of Vaš [33] on directly-finite Leavitt path algebras.

Finite-direct-injective modules

In this paper, we generalize the concept of direct-injective modules to fi nite-direct-injective modules. Various basic properties of these modules are studied. We show that the class of fi nite-direct-injective modules lies between the class of direct-injective modules and the class of simple-direct-injective modules. Also, we characterize semisimple artinian rings, V -rings and regular rings in terms of fi nite-direct-injective modules.

Koszul pairs and applications

Let R be a semisimple ring. A pair (A,C) is called almost-Koszul if A is a connected graded R -ring and C is a compatible connected graded R -coring. To an almost-Koszul pair one associates three chain complexes and three cochain complexes such that one of them is exact if and only if the others are so. In this situation (A,C) is said to be Koszul. One proves that a connected R -ring A is Koszul if and only if there is a connected R -coring C such that (A,C) is Koszul. This result allows us to investigate the Hochschild (co)homology of Koszul rings. We apply our method to show that the twisted tensor product of two Koszul rings is Koszul. More examples and applications of Koszul pairs, including a generalization of Fröberg Theorem [12], are discussed in the last part of the paper.

On Baer–Kaplansky Classes of Modules

We are interested in studying when the class of local modules is Baer–Kaplansky. We provide an example showing that even over a commutative semisimple ring R, we can find two non-isomorphic simple R-modules S1 and S2 such that the rings EndR(S1) and EndR(S2) are isomorphic. We show that over any ring R, the class of semisimple R-modules is Baer–Kaplansky if and only if so is the class of simple R-modules.

Rings all of whose right ideals are U-modules

ABSTRACTThe notion of a U-module was introduced and thoroughly investigated in [11] as a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper a right R-module M is called a U*-module if every submodule of M is a U-module, and a ring R is called a right U*-ring if RR is a U*-module. We show that M is a U*-module iff whenever A and B are submodules of M with A≅B and A∩B = 0, A⊕B is a semisimple summand of M; equivalently M = X⊕Y, where X is semisimple, Y is square-free, and X & Y are orthogonal. In particular, a ring R is a right U*-ring iff R is a direct product of a square-full semisimple artinian ring and a right square-free ring. Moreover, right U*-rings are shown to be directly-finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable-range 1, and hence is stably-finite. These results are non-trivial extensions of similar ones on rings all of whose right ideals are either quasi-continuous or auto-invariant.

Two acyclicizations of a complex and an explicit recollement

By assigning to each complex over a semi-simple ring two acyclicizations, we construct an explicit recollement for homotopy categories of a certain triangular matrix ring such that all the six triangle functors of the recollement preserve acyclic complexes.

PURE-INJECTIVITY FROM A DIFFERENT PERSPECTIVE

AbstractThe study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

On Right Orthogonal Classes and Cohomology Over Ding–Chen Rings

In this paper, we investigate the properties of right orthogonal modules of \({{\mathscr {C}}}\), where \({{\mathscr {C}}}\) is a class of left R-modules. As an application, we investigate the properties of right orthogonal modules of Ding injective left R-modules, and present various characterizations of semisimple and von Neumann regular rings and so on. Moreover, we also consider another cohomology, strong Tate cohomology, which connects the usual cohomology with the Ding cohomology.

On some classes of modules and characterization of rings

ABSTRACTAn abstract class of R-Mod is termed Additive Class if it is closed under taking submodules, homomorphic images and finite direct sums. In this paper we study the lattice properties of R-ad, the big lattice of all additive classes in R-mod for R any ring. Also, we study the lattice properties of R-ad when R is a Noetherian or semisimple ring. We describe a characterization of some rings through the additive classes and their properties. We study some properties of additive classes related with the associated linear filters.

FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS

In this paper we study some classes of rings which have a finite lattice of preradicals. We characterize commutative rings with this condition as finite representation type rings, i.e., artinian principal ideal rings. In general, it is easy to see that the lattice of preradicals of a left pure semisimple ring is a set, but it may be infinite. In fact, for a finite dimensional path algebra Λ over an algebraically closed field we prove that Λ-pr is finite if and only if its quiver is a disjoint union of finite quivers of type An; hence there are path algebras of finite representation type such that its lattice of preradicals is an infinite set. As an example, we describe the lattice of preradicals over Λ = kQ when Q is of type An and it has the canonical orientation