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On rings in which every ideal is the annihilator of an element

1968; American Mathematical Society; Volume: 19; Issue: 6 Linguagem: Inglês

10.1090/s0002-9939-1968-0234989-2

1088-6826

Cleon R. Yohe,

Commutative Algebra and Its Applications

Let R be a semisimple Artin ring (i.e. R is a direct sum of matrix rings over division rings). Then if L is a left ideal of R, L = Re where e is an idempotent and so L is exactly the left annihilator of the element 1 -e. We investigate the structure of rings having this property. If S is a subset of a ring R, let e(S) and r(S) denote the left and right annihilators of S. The notation eR(S) will be used when it is necessary to specify the ring R. DEFINITION. R is a left elemental annihilator ring (l.e.a.r.) if, whenever L is a left ideal of R, there exists an element aER such that L=e(a). A right elemental annihilator ring (r.e.a.r.) is defined analogously. Notice that if R is any ring (always assumed to have a unity element) and S is a subset of R, then e(S)=e(r(e(S))) and r(S) =r(e(r(S))). Hence if R is a l.e.a.r., and L is a left ideal, we have in particular that L==e(r(L)). If R is a r.e.a.r., and I a right ideal, then I=r(e(I)). We consider first the commutative case. It is known [1, Theorem 1.1] that if R is completely primary (local with nilpotent maximal ideal), then R has the property that every ideal is the annihilator of some subset of R if and only if R has a unique minimal ideal. Using the methods of Theorem I below, it is possible to derive from this the fact that if R is a commutative noetherian ring, then every ideal of R is the annihilator of some subset if and only if R is a direct sum of completely primary rings each of which has a unique minimal ideal. By imposing the more strenuous condition that R actually be an elemental annihilator ring, we obtain a similar result without the hypothesis of the chain condition.