It is well known that if M is a module with finite spanning dimension, then one can talk of Sd(K), the spanning dimension of K only when K is a supplement submodule in M. In this paper we extend this concept to general submodules and obtained some important results. We characterize the set of all supplement submodules of the module R/(x) over R where R is a Euclidean domain and x ∈ R. Moreover, it is proved that the number of distinct supplements in R/(x) is 2k and Sd(R/(x)) = k where k is the number of distinct nonassociate prime factors of x.

It is shown that a two-sided continuous ring with ascending chain condition on left annihilators is quasi-Frobenius.

The interesting question of the title was posed by J. Bergen and this note answers it in the negative. The main result characterizes rings which can be a finite union of proper right annihilators, and shows that any such commutative ring must have a total annihilator.

A prime Goldie ring K, in which each finitely generated left ideal is principal is the endomorphism ring E(F, A) of a free module A, of finite rank, over an Ore domain F. We determine necessary and sufficient conditions to insure that whenever K ≅ E(F, A) ≅ E(G, B) (with A and B free and finitely generated over domains F and G) then (F, A) is semi-linearly isomorphic to (G, B). We also show, by example, that it is possible for K ≅ E(F, A ) ≅ E(G, B), with F and G, not isomorphic.

If R is a commutative Noetherian ring and I is a nonzero ideal of R, it is known that R+I[x] is a Noetherian ring exactly when I is idempotent, and so, when R is a domain, I = R and R has identity. In this paper, the noncommutative analogues of these results, and the corresponding ones for power series rings, are proved. In the general case, the ideal I must satisfy the idempotent condition that TI = T for each right ideal T of R contained in I. It is also shown that when every ideal of R satisfies this condition, and when R satisfies the descending chain condition on right annihilators, R must be a finite direct sum of simple rings with identity.