Uniqueness properties of coprimary decompositions of modules over non-commutative rings are presented. In this paper, by making use of the technique employed, we shall prove uniqueness properties of coprimary decompositions. | Turk J Math 31 (2007) , 53 – 64. ¨ ITAK ˙ c TUB Uniqueness of Coprimary Decompositions M. Maani-Shirazi and P. F. Smith Abstract Uniqueness properties of coprimary decompositions of modules over non-commutative rings are presented. Key Words: Coprimary, decomposition, normal decomposition, prime ideal, left Noetherian ring, right Noetherian ring. 1. Introduction Throughout this paper, R is a ring (not necessarily commutative) with an identity element 1 = 0 and M is a non-zero unital left R-module. For any submodules N, L of M , we define (N : L) = {r ∈ R : rL ⊆ N }. Note that (N : L) is an ideal of R. Moreover, (N : L) = R if and only if L ⊆ N . Let N be a submodule of M and let A be an ideal of R; we set (N :M A) = {m ∈ M : Am ⊆ N }. Note that (N :M A) is a submodule of M . In this paper, by making use of the technique employed in [7], we shall prove uniqueness properties of coprimary decompositions. Note that, when R is a commutative Noetherian ring, M is coprimary if and only if M is secondary. It is well known that every non-zero injective module over a commutative Noetherian ring has a secondary representation (see [6]). By a similar method to that used in [6], we obtain the following result. For R non-commutative left and right Noetherian we show that if M is injective and if the zero ideal of R is a finite intersection of strongly primary ideals, then M has a coprimary decomposition. 53 MAANI SHIRAZI, SMITH 2. Coprimary Decompositions Definition. Given a prime ideal P of R, a non-zero module M is called P -coprimary if (i) (N : M ) ⊆ P for every proper submodule N of M , and (ii) P h ⊆ (0 : M ) for some positive integer h. Note that if M is P -coprimary, then P h ⊆ (0 : M ) ⊆ P for some positive integer h. M is called coprimary if it is P -coprimary for some prime ideal P of R. A non-zero module M has a coprimary decomposition if there exist a positive integer n and submodules Mi (1 ≤ i ≤ n) of M such that (i) M = M1 + · · · + Mn , and (ii) Mi is .
