Exercise 24, § 2

Let M be an A-module, I its injective envelope (II, p. 186, exerc. 18), a a two-sided ideal of A, Q the submodule of I formed by the $z \in I$ such that $az = \{0\}$; Q is naturally endowed with a $(A/a)$-module structure. Let N be the submodule of M formed by the $x \in M$ such that $ax = \{0\}$; if N is considered as a $(A/a)$-module, now prove that Q is isomorphic to an injective envelope of N.

25 a) For an A-module Q to be injective, it is enough that for every projective A-module P and every submodule P’ of P, every linear mapping from P’ into Q extends to a linear mapping from P into Q (use the fact that every A-module is a quotient of a projective A-module). b) For an A-module P to be projective, it is enough that for every injective A-module Q and every quotient module Q” of Q, every homomorphism from P into Q” is of the form φ ∘ u, where φ : Q → Q” is the canonical homomorphism and u is a homomorphism from P into Q (use the fact that every A-module is a submodule of an injective A-module).

26) For every $\mathbf{Z}$-module G and every integer $n > 0$, denote by $_n G$ the kernel of the endomorphism $x \mapsto nx$ of G. If $0 \to G' \to G \to G'' \to 0$ is an exact sequence, define a canonical homomorphism $d : _n G'' \to G'/nG'$ such that the sequence

is exact. If $_n G$ and $G/nG$ are finite, put

now prove that $_n G', _n G'', G'/nG', G''/nG''$ are finite and that one has $h_n(G) = h_n(G') + h_n(G'')$.