Exercise 12, § 2

(a) Let $A$ be a ring. Show that the following four properties are equivalent:

$(a)$ The right $A$-module $A,$ is coherent (Exercise 11). $(\beta)$ Every finitely presented right $A$-module is coherent. $(y)$ Every $A$-module $A^I_s$ (I an arbitrary set) is flat. $(\delta)$ Every product of flat left $A$-modules is flat.

(To prove that $(a)$ implies $(\beta)$, use Exercise 11(b). To see that $(y)$ implies $(a)$, use Proposition 13 of no. 11 and argue by reductio ad absurdum. To show that $(a)$ implies $(\delta)$, use Exercises 9.)

Such a ring is called right coherent and the concept of a left coherent ring is defined similarly.

(b) Show that every right Noetherian ring is right coherent. Give an example of a right Artinian ring which is not left coherent (cf. Algebra, Chapter VIII, § 2, Exercise 4).

(c) The ring of a non-discrete valuation of height 1 (Chapter VI) is coherent but contains ideals which are not coherent and admits (monogenous) quotient modules which are not pseudo-coherent.

(d) Show that, if $A$ is a right coherent ring, then, for every right $A$-module $E$, $\lambda(E) = -1$ or $\lambda(E) = 0$ or $\lambda(E) = +\infty$ (Exercise 6).

(e) Let $(A_\alpha, \phi_{\beta\alpha})$ be a direct system of rings whose indexing set is directed and let $A = \varprojlim A_\alpha$. Suppose that, for $a \leq \beta$, $A_a$ is a flat right $A_\alpha$-module. Show that, if the $A_a$ are right coherent, so is $A$. (Observe that $A$ is a flat $A_\alpha$-module for all $\alpha$ and that, if $E$ is a finitely generated submodule of $A_\alpha$, there exists an index $a$ and a finitely generated submodule $E_\alpha$ of $(A_\alpha)_a$ such that $E_\alpha \otimes_{A_\alpha} A$ is isomorphic to $E$.)

\* (f) Deduce from (e) that every polynomial ring (in any finite or infinite set of indeterminates) over a Noetherian commutative ring is coherent. Deduce from this that a quotient ring of a coherent ring is not necessarily coherent.*

(g) For A to be left coherent, it is necessary and sufficient that the left annihilator of every element of A be finitely generated and that the intersection of two finitely generated left ideals of A be finitely generated (use Exercise 6(f)).