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