03553nam a22005055i 4500001001800000003000900018005001700027007001500044008004100059020003700100024003500137050001200172072001600184072002300200082001500223100003100238245009900269264004000368300003300408336002600441337002600467338003600493347002400529490004100553505033800594520160300932650001702535650001302552650002302565650002102588650001802609650001702627650003602644650003802680650001302718700002702731710003402758773002002792776003602812830004102848856004802889912001402937999001902951952007702970978-3-7643-9990-0DE-He21320180115171805.0cr nn 008mamaa100301s2009 sz | s |||| 0|eng d a97837643999009978-3-7643-9990-07 a10.1007/978-3-7643-9990-02doi 4aQA251.5 7aPBF2bicssc 7aMAT0020102bisacsh04a512.462231 aKasch, Friedrich.eauthor.10aRegularity and Substructures of Homh[electronic resource] /cby Friedrich Kasch, Adolf Mader. 1aBasel :bBirkhàˆuser Basel,c2009. aXV, 164 p.bonline resource. atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aFrontiers in Mathematics,x1660-80460 aNotation and Background -- Regular Homomorphisms -- Indecomposable Modules -- Regularity in Modules -- Regularity in HomR(A, M) as a One-sided Module -- Relative Regularity: U-Regularity and Semiregularity -- Reg(A, M) and Other Substructures of Hom -- Regularity in Homomorphism Groups of Abelian Groups -- Regularity in Categories. aRegular rings were originally introduced by John von Neumann to clarify aspects of operator algebras ([33], [34], [9]). A continuous geometry is an indecomposable, continuous, complemented modular lattice that is not ?nite-dimensional ([8, page 155], [32, page V]). Von Neumann proved ([32, Theorem 14. 1, page 208], [8, page 162]): Every continuous geometry is isomorphic to the lattice of right ideals of some regular ring. The book of K. R. Goodearl ([14]) gives an extensive account of various types of regular rings and there exist several papers studying modules over regular rings ([27], [31], [15]). In abelian group theory the interest lay in determining those groups whose endomorphism rings were regular or had related properties ([11, Section 112], [29], [30], [12], [13], [24]). An interesting feature was introduced by Brown and McCoy ([4]) who showed that every ring contains a unique largest ideal, all of whose elements are regular elements of the ring. In all these studies it was clear that regularity was intimately related to direct sum decompositions. Ware and Zelmanowitz ([35], [37]) de?ned regularity in modules and studied the structure of regular modules. Nicholson ([26]) generalized the notion and theory of regular modules. In this purely algebraic monograph we study a generalization of regularity to the homomorphism group of two modules which was introduced by the ?rst author ([19]). Little background is needed and the text is accessible to students with an exposure to standard modern algebra. In the following, Risaringwith1,and A, M are right unital R-modules. 0aMathematics. 0aAlgebra. 0aAssociative rings. 0aRings (Algebra). 0aGroup theory.14aMathematics.24aAssociative Rings and Algebras.24aGroup Theory and Generalizations.24aAlgebra.1 aMader, Adolf.eauthor.2 aSpringerLink (Online service)0 tSpringer eBooks08iPrinted edition:z9783764399894 0aFrontiers in Mathematics,x1660-804640uhttp://dx.doi.org/10.1007/978-3-7643-9990-0 aZDB-2-SMA c373046d373046 001040708EBookaelibbelibd2018-01-15r2018-01-15w2018-01-15yEBOOK