Injectivity and Some of Its Generalizations

Gary F. Birkenmeier, Jae Keol Park and S. Tariq Rizvi
Additional contact information
Gary F. Birkenmeier: University of Louisiana at Lafayette, Department of Mathematics
Jae Keol Park: Busan National University, Department of Mathematics
S. Tariq Rizvi: The Ohio State University at Lima, Department of Mathematics

Chapter Chapter 2 in Extensions of Rings and Modules, 2013, pp 19-60 from Springer

Abstract: Abstract Conditions which generalize injective modules and which relate to this notion are presented in this chapter. The notions of extending, quasi-continuous and continuous modules are discussed and a number of their properties are included. It is known that direct summands of modules satisfying either of (C1), (C2) or (C3) conditions, of (quasi-)injective modules, or of (quasi-)continuous modules inherit these respective properties. On the other hand, these classes of modules are generally not closed under direct sums. One focus of this chapter is to discuss conditions which ensure that such classes of modules are also closed under direct sums. Applications of these notions, which include decomposition theorems, are also considered. As a natural generalization of the extending property, the notion of an FI-extending module (i.e., a module for which every fully invariant submodule is essential in a direct summand) is presented. A longstanding open problem is the precise characterization of when a direct sum of extending modules is extending. Using the FI-extending property, one can see that an arbitrary direct sum of extending modules satisfies at least the extending property for its fully invariant submodules without any additional conditions. The closely related notion of strongly FI-extending modules is also introduced. Properties of FI-extending and strongly FI-extending modules are discussed.

Keywords: Relatively injective; Quasi-injective; (C1); (C2); and (C3) conditions; Complement; Continuous; Quasi-continuous; Extending (CS); (Finitely generated) subfactor; Cyclic subfactor; V-module; Loewy length; σ[M]; Semiregular; Ker-injective; Densely nil; Directly finite; Purely infinite; Subisomorphic; Internal quasi-continuous hull; Essential homomorphism; Cancellative module; FI-extending; Regular ideal; Ideal essential; Ideal essential extension; Strongly FI-extending; Orthogonal modules (search for similar items in EconPapers)
Date: 2013
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-0-387-92716-9_2

Ordering information: This item can be ordered from
http://www.springer.com/9780387927169

DOI: 10.1007/978-0-387-92716-9_2

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().