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A subgroup H of a finite group G is called a weak second maximal subgroup of G if there exists a maximal subgroup M of G such that H is a maximal subgroup of M. Let m(G, H) denote the number of max...

Let ‎ $ ‎ \mathfrak{S}_p(_RM)$ be the lattice of all saturated submodules of an ‎ $ ‎ R ‎ $ ‎‏ -module ‎ $ ‎ M ‎ $ ‎‏ with respect to a prime ideal ‎ $ ‎ p ‎ $ ‎ of a ‎ commutative ring ‎‎‎‎ $ ‎ R ‎ $ ‎‏‏ . ‎ We examine the properties of the mappings ‎ $\eta:\mathfrak{S}_p(_RR)\rightarrow \mathfrak{S}_p(_RM)$ defined by $\eta(I)=S_p(IM)$ and $\theta:\mathfrak{S}_p(_RM)\rightarrow \mathfrak{S}_p(_RR)$ defined by $\theta(N)=(N:M)$ , in particular considering when these mappings are lattice homomorphism. It is proved that if ‎ $ ‎ M ‎ $ ‎‏ is a semisimple module or a projective module, then ‎ $ ‎ \eta ‎ $ ‎‏ is a lattice homomorphism. Also, if ‎ $ ‎ M ‎ $ ‎‏ is a faithful multiplication ‎ $ ‎ R ‎ $ ‎‏ -module, then ‎ $ ‎ \eta ‎ $ ‎‏ is a lattice epimorphism. In particular, if ‎ $ ‎ M ‎ $ ‎‏ is a finitely generated faithful multiplication ‎ $ ‎ R ‎ $ ‎‏ -module, then ‎ $ ‎ \eta ‎ $ ‎‏ is ‎ a ‎ lattice isomorphism and ‎ its inverse is $\theta$ . It is shown that if ‎ $ ‎ M ‎ $ ‎‏ is a distributive ‎‎‎‎ module over a semisimple ring ‎ $ ‎ R ‎ $ ‎ , then the lattice $\mathfrak{S}_p(_RM)$ forms a Boolean algebra and ‎ $ ‎ \eta ‎ $ ‎‏ is a Boolean algebra homomorphism. ‏

Our question is what ring R which all modules over R are determined, up to isomorphism, by their endomorphism rings? Examples of this ring are division ring and simple Artinian ring. Any semi simple ring does not satisfy this property. We construct a semi simple ring R but R is not a simple Artinian ring which all modules over R are determined, up to isomorphism, by their endomorphism rings.

Schur’s Lemma says that the endomorphism ring of a simple left R-module is a division ring. It plays a fundamental role to prove classical ring structure theorems like the Jacobson Density Theorem and the Wedderburn–Artin Theorem. We first define the endomorphism ring of simple left R-modules by their $$\Pi ^{0}_{1}$$ subsets and show that Schur’s Lemma is provable in $$\mathrm RCA_{0}$$ . A ring R is left primitive if there is a faithful simple left R-module and left semisimple if the left regular module $$_{R}R$$ is semisimple. The Jacobson Density Theorem and the Wedderburn-Artin Theorem characterize left primitive ring and left semisimple ring, respectively. We then study such theorems from the standpoint of reverse mathematics.

Let R be a finite ring. The zero-divisor graph of R, written as G(R), is a digraph with vertex set Z*(R), the set of all nonzero zero-divisor of R, and for x,y∈Z*(R) (needless distinct) there exist...

A module M is called ss-supplemented if every submodule U of M has a supplement V in M such that U(intersection) V is semisimple. It is shown that a finitely generated module M is ss-supplemented if and only if it is supplemented and Rad(M) (submodule) Soc(M). A module M is called strongly local if it is local and Rad(M) (submodule) Soc(M). Any direct sum of strongly local modules is ss-supplemented and coatomic. A ring R is semiperfect and Rad(R) (submodule) Soc( R R) if and only if every left R-module is ss-supplemented if and only if R R is a finite sum of strongly local submodules.

In this work the notion of left (right) self-injective ring is generalized. We consider rings that are direct sum of injective module and semisimple module as a left (respectively, right) module above itself. We call such rings left (right) semi-injective and research their properties with the help of two-sided Peirce decomposition of the ring. The paper contains the description of left noetherian left semi-injective rings. It is proved that any such ring is a direct product of (two-sided) self-injective ring and several quotient rings (of special kind) of rings of upper triangular matrices over skew fields. From this description it follows that for left semi-injective rings we have the analogue of the classical result for self-injective rings. Namely, if a ring is left noetherian and left semi-injective then this ring is also right semi-injective and two-sided artinian.

Modules are a generalization of the vector spaces of linear algebra in which the “scalars” are allowed to be from a ring with identity, rather than a field. In module theory there is a concept about projective module, i.e. a module over ring R in which it is projective module relative to all modules over ring R. Next, there is the fact that every module over ring R is projective module relative to all semisimple modules over ring R. If P is a module over ring R which it’s projective relative only to all semisimple modules over ring R, then P is called p-poor module. In the discussion of the projective module, there is a lemma associated with the equivalence of two modules K1 and K2 provided that there are two projective modules P1 and P2 such that is isomorphic to . That lemma is known as Schanuel’s lemma in projective modules. Because the p-poor module is a special case of the projective module, then in this paper will be discussed about Schanuel’s lemma in p-poor modules

Jayaram and Tekir defined an R-module M, R is a commutative ring, to be “von Neumann regular” if for each m∈M, there exists an a∈R such that Rm=aM=a2M. Previously, Fieldhouse called M “regular” if ...

Let p1,p2,…,pr,q be distinct primes. Let m=∏i=1rpiαi, 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 c...

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 Communi...

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.

We carry out a study of rings $R$ for which $\operatorname{Hom}_R(M,N)\neq 0$ for all nonzero $ N\leq M_R$. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Köthe ring $R$ is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when $R$ is retractable.

If $M$ is a simple module over a ring $R$ then, by the Schur's lemma, the endomorphism ring of $M$ is a division ring. However, the converse of this result does not hold in general, even when $R$ is artinian. In this short note, we consider perfect rings for which the converse assertion is true, and we show that these rings are exactly the primary decomposable ones.

Bu yüksek lisans tezinde büyük projektif modül kavramı tanımlandı ve bu kavramla ilgili birtakım özellikler incelendi. Bu çalışmada tüm modüller birimli halka üzerinde olup üniter modüllerdir. M bir R-modül ve K, M'nin bir büyük alt modülü olmak üzere eğer M/K büyük M-projektif ise K=M olur. M bir büyük projektif R-modül olsun. N bir R-modül olmak üzere eğer N'den M'ye çekirdeği N'de büyük olan en az bir R-modül epimorfizması tanımlanabilirse bu durumda M=0 olur. M bir R-modül olmak üzere büyük M-projektif modüllerin direkt toplamının da büyük M-projektif olduğu gösterildi. Ayrıca büyük M-projektif bir modülün her direkt toplam teriminin de büyük M-projektif olduğu gösterildi. M bir büyük N-projektif R-modül ve K, N'nin bir alt modülü olsun. Bu durumda M büyük N/K-projektif olur. M bir büyük N-projektif R modül ve K, N'nin bir büyük alt modülü olsun. Bu durumda M büyük K-projektif olur. M ve N iki R-modül olsun. Bu durumda M modülünün N'nin her kopyalarının direkt toplamına göre büyük projektif olması için gerek ve yeter şart her N-üretilmiş L modülü için M'nin büyük L-projektif olmasıdır. M bir R-modül olsun. Bu durumda her R-modülün büyük M-projektif olması için gerek ve yeter şart M'nin yarı basit olmasıdır. Her R-modülün büyük M-projektif olması için gerek ve yeter şart M'nin temelinin M'ye eşit olmasıdır. Yine her R-modülün büyük M-projektif olması için gerek ve yeter şart her R-modülün M-projektif (M-injektif) olmasıdır. P bir R-modül olsun. Eğer P büyük P-projektif ise P'ye bir kendi kendine büyük projektif R-modül denir. Tezin sonunda P kendi kendine büyük projektif modül ve K, P'nin tamamen değişmez bir alt modülü olmak üzere P/K'nın kendi kendine büyük projektif olduğu gösterildi. In this thesis, essential projective modules are defined and some properties of these modules are investigated. All rings have identities and all modules are unital modules in this work. Let M be an R-module and K be an essential submodule of M. If M/K is essential M-projective, then K=M. Let M be an essential projective R-module. If it can be defined an R-module epimorphism from N to M with essential kernel for an R module N, then M=0. For an R-module M, it is proved that the direct sum of essential M-projective modules is essential M-projective. It is also proved that every direct summand of an essential M-projective module is essential M-projective. Let M be an essential N-projective R-module and K be a submodule of N. Then M is essential N/K projective. Let M be an essential N-projective R-module and K be an essential submodule of N. Then M is essential K-projective. Let M and N be R-modules. Then M is essential projective according to every direct sum of copies of N if and only if M is essential L-projective for every N-generated module L. Let M be an R-module. Then every R-module is essential M-projective if and only if M is semisimple. Every R-module is essential M-projective if and only if the socle of M equal to M. Also every R-module is essential M-projective if and only if every R-module is M-projective (M-injective). Let P be an R-module. If P is essential P projective, then P is called a self essential projective R-module. At the end of this thesis, for a self essential projective R-module P and a fully invariant submodule K of P, it is proved that P/K is self essential projective.

Let $$f:A\rightarrow B$$ be a homomorphism of commutative rings and let J be an ideal of B. The amalgamation of A with B along J with respect to f is the subring of $$A\times B$$ given by $$A\bowtie ^fJ=\{(a,f(a)+j)\mid a\in A, \, j\in J\}$$ . In this paper, we give some characterizations for the amalgamation construction to be a quasi-Frobenius ring.

WOS: 000463698900056 In this paper, we provide various properties of GE and GEE-modules, a new variation of injective modules. We call M a GE-module if it has a g-supplement in every extension N and, we call also M a GEE-module if it has ample g-supplements in every extension N. In particular, we prove that every semisimple module is a GE-module. We show that a module M is a GEE-module if and only if every submodule is a GE-module. We study the structure of GE and GEE-modules over Dedekind domains. Over Dedekind domains the class of GE-modules lies between WS-coinjective modules and Zoschinger's modules with the property (E). We also prove that, if a ring R is a local Dedekind domain, an R-module M is a GE-module if and only if M congruent to (R*)(n) circle plus K circle plus N, where R* is the completion of R, K is injective and N is a bounded module.

For a semisimple module M over a ring R with R / J ( R ) Boolean, every endomorphism of M is a sum of an idempotent endomorphism and a locally nilpotent endomorphism. As a consequence, it is proved that, for a vector space V over a division ring D, every linear transformation of V is a sum of an idempotent linear transformation and a locally nilpotent linear transformation if and only if D ≅ F 2 . This can be seen as an answer to the “local” version of a question raised by Breaz et al. in [1] on nil-cleanness of the ring of linear transformations of an infinite dimensional vector space.

We say that an R-module M is virtually semisimple if each submodule of M is isomorphic to a direct summand of M. A nonzero indecomposable virtually semisimple module is then called a virtually simple module. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules . Our study provides several natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring R: (i) Every finitely generated left (right) R-module is virtually semisimple; (ii) Every finitely generated left (right) R-module is a direct sum of virtually simple R-modules; (iii) $R\cong {\prod }_{i = 1}^{k} M_{n_{i}}(D_{i})$ where k,n 1,…,n k ∈ ℕ and each D i is a principal ideal V-domain; and (iv) Every nonzero finitely generated left R-module can be written uniquely (up to isomorphism and order of the factors) in the form R m 1 ⊕… ⊕ R m k where each R m i is either a simple R-module or a virtually simple direct summand of R.

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.

By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them {\it virtually semisimple modules}. A module $_RM$ is called {\it completely virtually semisimple} if each submodules of $M$ is a virtually semisimple module. A ring $R$ is then called {\it left} ({\it completely}) {\it virtually semisimple} if $_RR$ is a left (compleatly) virtually semisimple $R$-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring $R$ is left completely virtually semisimple if and only if $R \cong \prod _{i=1}^ k M_{n_i}(D_i)$ where $k, n_1, ...,n_k\in \Bbb{N}$ and each $D_i$ is a principal left ideal domain. Moreover, the integers $k,~ n_1, ...,n_k$ and the principal left ideal domains $D_1, ...,D_k$ are uniquely determined (up to isomorphism) by $R$.