Abstract :
We study the following properties about primary decomposition over a Noetherian ring R: (1) For finitely generated modules Nsubset of or equal toM and a given subset X={P1,P2,…,Pr}subset of or equal toAss(M/N), we define an X-primary component of Nsubset of with not equal toM to be an intersection Q1∩Q2∩cdots, three dots, centered∩Qr for some Pi-primary components Qi of Nsubset of or equal toM and we study the maximal X-primary components of Nsubset of or equal toM; (2) We give a proof of the ‘linear growth’ property of Ext and Tor, which says that for finitely generated modules N and M, any fixed ideals I1,I2,…,It of R and any fixed integer image, there exists a image such that for any image there exists a primary decomposition of 0 in image (or 0 in image) such that every P-primary component Q of that primary decomposition contains image (or image), where image.
