Exercise 28, § 2

A finitely generated A-module E is called strongly Laskerian if every submodule of E admits a primary decomposition in E all of whose elements are strongly primary (Exercise 27).

(a) Show that for a finitely generated A-module E to be strongly Laskerian, it is necessary and sufficient that it satisfy axiom (LA_{II}) of Exercise 23 and the axiom:

(LA_{III}) For every sequence (b_k) of ideals of A and every submodule F of E, the increasing sequence of submodules F: $(b_1 b_2 \ldots b_k)$ is stationary.

(To show that the conditions are necessary, use Exercise 27 (d). To see that they are sufficient, prove first that (LA$_{III}$) implies (LA$_I$) and then that it implies that every primary ideal is of finite exponent, using Exercise 27 (a).)

(b) Show that the ring A defined in Exercise 23 (c) satisfies (LA_{III}).

(c) Let K be a field, V an infinite dimensional vector K-space and $A = K \oplus V$ the ring whose multiplication is defined by

Show that in A every ideal $\#A$ is strongly primary, although A is not Noetherian.

(d) Let E be a finitely generated A-module and $E' \subset E$ a finitely generated submodule. For E to be strongly Laskerian, it is necessary and sufficient that E' and $E/E'$ be so. In particular, if A is a strongly Laskerian ring, every finitely generated A-module is strongly Laskerian. If E is a strongly Laskerian A-module, S^{-1}E is a strongly Laskerian S^{-1}A-module for every multiplicative subset S of A.

(e) Generalize Exercises 3 and 4 to strongly Laskerian modules.