Definition

We say that an A-module I is a Matlis A-module if it is injective, if m_A is its unique associated prime ideal, and if the κ_A-vector space Hom_A(κ_A, I) is of dimension 1.

Let e : κ_A → I be an injective envelope of κ_A (A, X, p. 20, th. 2). The A-module I is a Matlis module, and every Matlis module is isomorphic to I (No. 2, cor. of prop. 3). If A is a discrete valuation ring, with field of fractions K, the A-module K/A is a Matlis module (No. 1, example). If A is an artinian local ring, the A-module A is a Matlis module if and only if A is a Gorenstein ring (§ 3, No. 7, lemma 1).

Let I be a Matlis A-module. For every integer n ≥ 0, let I_n denote the sub-A-module of I consisting of the elements annihilated by m_A^n. By prop. 2 of No. 2, the A-module I is the union of the I_n, the A-module I_1 is of length 1 (that is to say isomorphic to κ_A) and the A-module I is an injective envelope of I_1; moreover, the canonical homomorphism of graded gr(A)-modules

is an isomorphism. By prop. 3 of No. 2, the A-module structure of I extends to a unique $\widehat{A}$-module structure, and the canonical homomorphism $\widehat{A} \to \operatorname{End}_A(I)$ is bijective.