It is shown that:
(1) Let R be a simple right Noetherian ring, then the following conditions are equivalent: (i) R is a right SI ring; (ii) Every cyclic singular right R - module is pseudo - injective. (2) Let R be a right artinian ring such that every finite generated right R - module is a direct sum of a projective module and a pseudo - injective module. Then: (i) R/Soc(RR ) is a semisimple artinian ring; (ii) J (R) ⊂ Soc(RR ); (iii) J 2 (R) = 0. (3) Let R be a ring with condition (∗ ),...
Commutative algebra is the theory of commutative rings and their modules.
Although it is an interesting theory in itself, it is generally seen as a tool for
geometry and number theory. This is my point of view. In this book I try
to organize and present a cohesive set of methods in commutative algebra, for
use in geometry. As indicated in the title, I maintain throughout the text a
view towards complex projective geometry.
In many recent algebraic geometry books, commutative algebra is often
treated as a poor relation. One occasionally refers to it, but only reluctantly.
Let (R, m) be a commutative Noetherian local ring the maximal ideal and A an Artinian R-module with Ndim A = d. For each system of parameters x (x1 , ..., xd ) of A, we denote by e(x, A) the multipility of A with respect to x. Let n (n1 , n2 , ..., nd ) be a d-tuple of positive integers. The paper concerns to the function d-variables I(x(n); A) :=