Summand Intersection Property Research Articles

Co-retractable Modules and Semi-Simplicity Property

Co-retractable modules appeared in 2006 by Clark, Lomp and Vanaja. In this article, we present and study the relationships between two known concepts namely semi-simple modules and Co-retractable module. Our concern in this article is to develop some properties of Co-retractable module and related to other classes of modules such as multiplication module, indecomposable module and projective module. We prove that if M is a multiplication R-module with annR(M) is a prime ideal and the Socal of M is a direct summand of M, this means M is a retractable module. Also, if M is an indecomposable multiplication R-module with annR(M) is a prime ideal of Noetherian Ring R and every direct sum of modules with summand intersection property has summand sum property, then M is a co-retractable module. Finally, if M is a simple quasi-Dedekind module over the Artinian Ring R with rad(R) = 0 and annR(M) is a prime ideal of R, then M is a co-retractable module.

Summand intersection property on the class of exact submodules

A module M is said to have the SIP if intersection of each pair of direct summands is also a direct summand of M. In this article, we define a module M to have the SIPr if and only if intersection of each pair of exact direct summands is also a direct summand of M where r is a left exact preradical for the category of right modules. We investigate structural properties of SIPr-modules and locate the implications between the other summand intersection properties. We deal with decomposition theory as well as direct summands of SIPr-modules. We provide examples by looking at special left exact preradicals.

Generalized SSP-modules

We say an R-module M has the generalized summand sum property (GSSP), if the sum of any two direct summands is isomorphic to a direct summand of M. This is dual notion to the generalized summand intersection property of modules, and also the class of modules having the GSSP is a proper generalization of the class of modules having the SSP. In this note, the characterization of this property over rings and modules is investigated.

Generalized SIP-Modules

We say an $R$-module $M$ has the generalized summand intersection property (briefly $GSIP$), if the intersection of any two direct summands is isomorphic to a direct summand. This is a generalization of SIP modules. In this note, the characterization of this property over rings and modules is investigated and some useful propositions obtained in SIP modules are generalized to GSIP modules.

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.

Generalizations of supplemented lattices

Some generalizations of the concept of a supplemented lattice, namely a soc-supplemented-lattice, soc-amply-supplemented-lattice, soc-weak-supplemented-lattice, soc-⊕-supplemented-lattice and completely soc-⊕-supplemented-lattice are introduced. Various results are proved to show the relationship between these lattices. We have also proved that, if L is a soc-⊕-supplemented-lattice satisfying the summand intersection property (SIP), then L is a completely soc-⊕-supplemented-lattice.

STRONGLY T-EXTENDING MODULES AND STRONGLY T-BAER MODULES

In this paper, we introduce the notions of strongly t-extending and strongly t-Baer modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that, while a direct summand of a strongly t-extending module inherits the property, direct sums of strongly t-extending modules do not. Moreover, when a direct sum of strongly t-extending modules is strongly t-extending, is investigated. Also, it is proved that every strongly t-extending module has strongly summand intersection property and densely co-Hopfian property.

S.Baer and s.Rickart Modules

In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for the s.Baer modules, and we provide examples of s.Baer torsion modules and modules with a nonzero s.Baer radical. The other principal interest of this paper is to provide explicit connections between s.Baer modules and projective modules. Among other results, we show that every s.Baer module is an essential extension of a projective module. Additionally, we prove, with limited and natural assumptions, that in a generalized triangular matrix ring every s.Baer submodule of the ring is projective. As an application, we show that every prime ring with a minimal right ideal has the strong summand intersection property. Numerous examples are provided to illustrate, motivate, and delimit the theory.

On Strongly Extending Modules

The purpose of this paper is to introduce the concept of strongly extending modules which are particular subclass of the class of extending modules, and study some basic properties of this new class of modules. A module M is called strongly extending if each submodule of M is essential in a fully invariant direct summand of M. In this paper we examine the behavior of the class of strongly extending modules with respect to the preservation of this property in direct summands and direct sums and give some proper- ties of these modules, for instance, strongly summand intersection property and weakly co-Hopflan property. Also such modules are characterized over commutative Dedekind domains.

On Non-M-Cosingular Completely ⊕-Supplemented Modules

In this paper, it is shown that any non- $M$ -cosingular $\oplus$ -supplemented module $M$ is $(D_3)$ if and only if $M$ has the summand intersection property. Let $N\in\sigma[M]$ be any module such that ${\overline { Z}}_M(N)$ has a coclosure in $N$ . Then we prove that $N$ is (completely) $\oplus$ -supplemented if and only if $N={\overline { Z}}_M^2(N) \oplus K$ for some submodule $K$ of $N$ such that ${\overline { Z}}_M^2(N)$ and $K$ both are (completely) $\oplus$ -supplemented.

On Summand Ideals in Commutative Reduced Rings#

Summand intersection property (SIP) and summand sum property (SSP) of modules are studied in Garcia [Garcia, J. L. (1989). Properties of direct summands of modules. Comm. Algebra 17(1):73–92] and Wilson [Wilson (1986). Modules with the summand intersection property. Comm. Algebra 14(1):21–38] respectively, and these properties for C(X) are studied in Azarpanah [Azarpanah, F. (1999). Sum and intersection of summand ideals in C(X). Comm. Algebra 27(11): 5549–5560]. In this paper we give some topological characterizations of these properties in the case of a commutative reduced ring R. It is shown that R has SIP if and only if every intersection of clopen subsets of Spec(R) has a closed interior. This characterization shows that R has SIP whenever Spec(R) is locally connected or extremally disconnected. It is also shown that R has SSP if and only if Spec(R) has only finitely many components. If 𝒮 is a strong subspace of Spec(R), we can replace 𝒮 with Spec(R) and all of the above results remain valid. In semiprimitive Gelfand rings, SIP and SSP are equivalent.

Matrix Rings with Summand Intersection Property

A ring R has right SIP (SSP) if the intersection (sum) of two direct summands of R is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of R by M has SIP if and only if R has SIP and (1 − e)Me = 0 for every idempotent e in R. Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.

Sum and intersection of summand ideals in C(X)

Summand sum property (SSP) and summand intersection property (SIP) of modules are studied in [8] and [15] respectively. In this paper we give some topological characterizations of these properties in C(X). It is shown that the ring C(X) has SIPif and only if every intersection of closed-open subsets of Xhas a closed interior. This characterization then shows that for a large class of topological spaces, such as locally connected spaces and extremally disconnected spaces, the ring C(X) has SIP. It is also shown that C(X) has SSPif and only if the space Xhas only finitely many components. Finally, using summand ideals of C(X), we will give several algebraic characterizations of some disconnected spaces.

Modules with the quasi-summand intersection property

Given a torsion-free abelian group G, a subgroup A of G is said to be a quasi-summand of G if nG ≤ A ⊕ B ≤ G for some subgroup B of G and some positive integer n. If the intersection of any two quasi-summands of G is a quasi-summand, then G is said to have the quasi-summand intersection property. This is a generalisation of the summand intersection property of L. Fuchs. In this note, we give a complete characterisation of the torsion-free abelian groups (in fact, torsion-free modules over torsion-free rings) with the quasi-summand intersection property. It is shown that such a characterisation cannot be given via endomorphism rings alone but must involve the way in which the endomorphism ring acts on the underlying group. For torsion-free groups G of finite rank without proper fully invariant quasi-summands however, the structure of its quasi-endomorphism ring QE(G) suffices: G has the quasi-summand intersection property if and only if the ring QE(G) is simple or else G is strongly indecomposable.

A characterization of modules with the summand intersection property

A characterization of modules with the summand intersection property

Modules with the summand intersection property

(1989). Modules with the summand intersection property. Communications in Algebra: Vol. 17, No. 1, pp. 135-148.

Modules with the summand intersection property

(1986). Modules with the summand intersection property. Communications in Algebra: Vol. 14, No. 1, pp. 21-38.