< jats: p> This volume provides a clear and self-contained introduction to important results in
the theory of rings and modules. Assuming only the mathematical background provided by …

… as defined by Serre and prove that over regular local rings of dim < 5, given two modules M,
… multiplicity in a more general set up. Finally we extend Serre's result from pairs of modules …

… If R is a ring and X is a submodule of an R-module M, the ideal |re R | rh4 g X) will be … that
every multiplication module over a local ring is cyclic. Let R be a local ring with maximal ideal …

… Some remarks on finite modules are stated in § 2. … multiplicity of a primary ideal, which is
generated by a system of parameters; in a local ring to that of a defining ideal in a finite module …

… In this work all rings are commutative and identity and all modules are finitely generated. We
study intersection multiplicities of modules over Cohen-Macauley and Gorenstein rings R of …

… In this paper we study both problems over a complete ramified regular local ring R with
maximal ideal m such that R/m is algebraically closed. (For the multiplicity problem without any …

… .where G(Mi) is the associated graded module. This allows also to give a treatment of tile
basic theory of multiplicities which is, as we hope, more satisfying than the existing approaches. …

… In this section we shall show how the multiplicity symbol changes when we replace a ring
by an integral extension. First we prove a result for lengths, which is in fact the extension …

… In this paper we will replace the single Buchsbaum-Rim multiplicity e(Q \M \ E) by a sequence
of multiplicities et(Q\M\E) defined for / = 0,1, ...,m—1. These are derived by considering the …

… The classes of rings we investigate include those that are determined by ideals … and
multiplicities to Betti numbers of semidualizing modules, starting with a general result for modules of …