SOME PROPERTIES OF NOETHERIAN LOCAL RINGS WHICH IDEALS ARE DIRECT SUMS OF CYCLIC MODULES

• Main Author: Rizqi Musa, Muhammad
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/39061
• Institution: Institut Teknologi Bandung

Direct sums of cyclic modules is one of algebraic structure that have important role in Algebra and its applications. One of the most important result on this topic was found by Gottfried Kothe in 1935. Kothe introduces a ring which every modules over it are direct sums of cyclic modules. In this book, we study about commutative rings R which ideals are direct sums of cyclic modules. In the case Noetherian local ring (R;M), the following statements are equivalent: (1) Every ideal of R are direct sums of cyclic modules; (2) M = Ln i=1 Rwi where at most two of the summand are not simple; (3) There exists n such that every ideal of R is a direct sums of at most n cyclic modules; and (4) Every ideal of R is a summand of a direct sums of cyclic modules. Wether the same is true if not every ideal of R is a direct sums of cyclic modules?. In the case Noetherian local ring (R;M), the following statements are equivalent: (1) Every prime ideal of R is a direct sums of cyclic modules; (2) M = Ln i=1 Rwi and R= Ann(wi) is a principal ideal ring for 1 i n; (3) There exists n such that every prime ideal of R is a direct sums of at most n cyclic modules; and (4) Every prime ideal of R is a summand of a direct sums of cyclic modules.