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A countable dimensional self-injective algebra is Artinian. There is an application to self-injective twisted group algebras. It has been known for some time that a countable self-injective ring is semilocal (see for example [8]). In this paper we show that such a ring is in fact quasi-Frobenius. Special cases of this result have been proved previously, for example if the ring is also regular [3] or if it is a group algebra [8]. My thanks to Ken Louden for his help in the preparation of this paper. Unless stated otherwise, all rings are associative with a unity. If S is a subset of a ring R, we denote its left annihilator in R by lR(S). Theorem 1 (Faith [1]). A ring is quasi-Frobenius if it is right self-injective and satisfies the descending chain condition on left annihilators. Proposition 2. Let R be a subring of S. Suppose that Ss is injective, RS is flat and SR is free. Then RR is injective. Proof. The proof is left to the reader. Theorem 3. Every countable subring of a quasi-Frobenius ring is contained in a countable quasi-Frobenius subring. Conversely, if every countable subring of a ring is contained in a quasi-Frobenius subring, then the ring is quasi-Frobenius. Proof. Suppose first that F is a quasi-Frobenius ring and A is a countable subring. We construct a sequence of subrings A = R0 e Rx E R2 C ■ • • C T inductively as follows. Given Rk, consider all «-tuples {a,, . . . , an) of elements of Rk as « ranges over the positive integers. If an E ax T + • • • + a„-\T choose xx, x2, . . . , xn_x E T so that an = axxx + ■ ■ ■ + an_xxn_x. If an G axT + • • • + an_xT, choose xn G T so that xna¡ = 0, ; = 1, 2, . . . , n — 1, and x„an =£ 0. Now do the same for the left ideal generated by ax, a2, . . . , an_x. Let Rk + X be the subring of T generated by Rk and all the x's obtained. Let R = U " i^,Clearly A E R and R is a countable subring, so Received by the editors June 23, 1976. AMS (A/OS) subject classifications (1970). Primary 16A52; Secondary 16A26.

A countable self-injective ring is quasi-Frobenius