Extensions of Rings and Modules by Gary F. Birkenmeier

By Gary F. Birkenmeier

The concentration of this monograph is the examine of jewelry and modules that have a wealthy provide of direct summands with recognize to varied extensions. the 1st 4 chapters of the publication talk about earrings and modules which generalize injectivity (e.g., extending modules), or for which convinced annihilators turn into direct summands (e.g., Baer rings). Ring extensions comparable to matrix, polynomial, staff ring, and crucial extensions of jewelry from the aforementioned periods are thought of within the subsequent 3 chapters. A thought of ring and module hulls relative to a selected type of jewelry or modules is brought and built within the following chapters. whereas purposes of the consequences offered are available during the booklet, the ultimate bankruptcy commonly includes purposes to algebra and sensible research. those comprise acquiring characterizations of earrings of quotients as direct items of top jewelry and outlines of yes C*-algebras through (quasi-)Baer jewelry.

Extensions of jewelry and Modules introduces for the 1st time in ebook form:

* Baer, quasi-Baer, and Rickart modules
* the speculation of generalized triangular matrix jewelry through units of triangulating idempotents
* A dialogue of crucial overrings that aren't jewelry of quotients of a base ring and Osofsky's examine at the self-injectivity of the injective hull of a ring
* purposes of the idea of quasi-Baer earrings to C*-algebras

Each component to the e-book is enriched with examples and routines which make this monograph invaluable not just for specialists but in addition as a textual content for complicated graduate classes. ancient notes seem on the finish of every bankruptcy, and an inventory of Open difficulties and Questions is equipped to stimulate extra study during this region.

With over four hundred references, Extensions of jewelry and Modules should be of curiosity to researchers in algebra and research and to complex graduate scholars in mathematics.

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Example text

By ( ), L/A is extending. Hence, every finitely generated subfactor of M is extending. 19, udim(M) is finite, and hence Soc(M) is semisimple Artinian. Similarly, udim(M/Soc(M)) is finite. So S/Soc(M) = Soc(M/Soc(M)) is semisimple Artinian. Therefore, S is a module of finite composition length. Because H = M/S is Noetherian by the preceding argument, M is Noetherian. 23 and we assume that every finitely generated R-module is extending, then the factor ring R/S has zero right socle and it is a right Noetherian, right V-ring (recall that a ring A is called a right V-ring if AA is a V-module, so any right ideal of A is an intersection of maximal right ideals).

In this chapter, we introduce various conditions which are related to extending or injective modules. It is known that direct summands of modules satisfying either of (C1 ), (C2 ) or (C3 ) conditions, or 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 closed under direct sums.

N. 7 immediately yields the result. 3 Let M be a module and n a positive integer. Then M (n) is quasiinjective if and only if M is quasi-injective. In the next result, we see that a direct summand of a quasi-continuous module is always relatively injective to all other direct summands. 4 Let M = i∈Λ Mi . If M is quasi-continuous, then each Mi is quasicontinuous and Mj -injective for all j = i. 2 Internal Quasi-continuous Hulls and Decompositions 35 Proof Let M = ⊕i∈Λ Mi be quasi-continuous.

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