Commutative algebra - Department of Mathematical Sciences - old ...

92 7. MODULES OF FINITE LENGTH
7.5.5. Corollary. Let (R, P ) be a local artinian ring and k(P ) ⊂ E an injective
envelope. There is an isomorphism
11E : R → HomR(E, E)
7.5.6. Proposition. Let (R, P ) be a local artinian ring and k(P ) ⊂ E an injective
envelope. Let M → E ′ be an injective envelope of a finite module. Then
(1) HomR(k(P ), M) HomR(k(P ), E ′ )
(2) E ′ E n , where n = ℓR(HomR(k(P ), M).
(3) ℓR(E ′ ) = ℓR(HomR(k(P ), M) ℓR(E).
Proof. (1) A homomorphism f : k(P ) → E ′ has Im f ⊂ M since the extension
is essential. (2) By induction on length of M follows that E ′ E n , where n is
determined by (1) and 7.5.2. (3) This follows from (2).
7.5.7. Proposition. Let (R, P ) be a local artinian ring and k(P ) ⊂ E an injective
envelope. The following are equivalent.
(1) R is injective.
(2) R E.
(3) HomR(k(P ), R) HomR(k(P ), k(P )) k(P ).
Proof. (1) ⇒ (2): By 7.5.6 R E n and by 7.5.3 n = 1. (2) ⇒ (3): Immediate
from 7.5.6. (3) ⇒ (1): Let R ⊂ E ′ be an injective envelope. By 7.5.6 R = E ′ .
7.5.8. Definition. A ring satisfying the conditions 7.5.7 is a local artinian Gorenstein
ring.
7.5.9. Example. Let R be a principal ideal domain and p ∈ R an irreducible
element. Then R/(p n ) is a local artinian Gorenstein ring.
7.5.10. Exercise. (1) Let p be a prime number. Show that Z/(p k ) is a local artinian
Gorenstein ring.