Finitely embedded modules over Noetherian rings

Abstract

A right R -module is finitely embedded if it has finitely generated essential socle (see, e.g., Vamos [4]). In [1] Jategaonkar settled the Jacobson conjecture for left and right fully bounded noetherian rings by showing that every finitely generated finitely embedded module is artinian. It is an open question whether or not this property holds for an arbitrary left and right noetherian ring (though it is well known that right noetherian is not enough). We prove here that if R is left and right noetherian and M is a projective finitely generated finitely embedded module over R, then M is artinian. This result can be extended to cover the case where M is an arbitrary finitely embedded submodule of a finitely generated free R-module. The proof of this and related results will appear elsewhere. Even the case M = R seems to be new and in this case we can obtain the more general THEOREM. If R is a left and right noetherian ring and if the right socle of R is either left or right essential, then R is artinian.We note that there exist right noetherian rings with right essential right socle which are not right artinian (see, e.g., [2] ).All rings have identity and modules are unitary. l A (X) and r A (X) denote, respectively, the left and right annihilators of Xin the ring 4.We require the following result of T. H. Lenagan [3] which is adapted to suit our purpose in the form of We also need the following lemma whose proof is straightforward.

A Decomposition Theorem for Noetherian orders in Artinian Rings

“…The result announced in [3] follows easily from n v ...,n k eN such that RN = Rn l + ...+Rn k . Thus There is then a right i?-module embedding:…”

“…This gives S 2 (CT) s (ST) 2 . Since CT eJ there exists (^ e / such that S 3 C x s (ST)3 . Continuing in this way gives eventually S P B £ (ST) P = 0 for some BeJ.lt follows that S p ^ A.THEOREM 9.…”

A counter example to a conjecture of Johns

“…It is well known that over a commutative ring, every Noetherian module with essential socle is Artinian. Although this is not true for arbitrary right Noetherian rings, some positive results have been obtained by Ginn and Moss [6] when the ring is right and left Noetherian. Johns [7,Theorem 1], by using a result of Kurshan [8,Theorem 3.3], showed that a right Noetherian ring is right Artinian, provided that every right ideal is a right annihilator.…”

A Counter Example to a Conjecture of Johns

“…Although this is not true for arbitrary right Noetherian rings, some positive results have been obtained by Ginn and Moss [6] when the ring is right and left Noetherian. Johns [7,Theorem 1], by using a result of Kurshan [8,Theorem 3.3], showed that a right Noetherian ring is right Artinian, provided that every right ideal is a right annihilator.…”