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For a certain class of semigroup algebras kA, we show that the ring of all differential operators on kA is Morita equivalent to the first Weyl algebra A1. Let k be a field, and K a commutative k-algebra. We denote by D0(K, K) the set of K-linear maps K -+ K, and if p > 1 then a k-linear map f: K -K belongs to Dp(K, K) provided the map [f, r] defined by [f, r](s) = f(rs) rf(s) for s E K belongs to DP-,(K, K) for all r e K. The set D(K) = Up>ODP(K, K) forms a subring of Endk (K) called the ring of differential operators on K. We consider K as a left D(K)-module where f * r = f(r) for all f e D(K), r E K. In this note we show that for certain subalgebras K of k[t], where k is a field of characteristic zero and t an indeterminate, the ring D(K) is Morita equivalent to the Weyl algebra A1. If A is a subsemigroup of the set of nonnegative integers N, we let kA be the semigroup algebra on A, which we regard as the subalgebra of k[t] spanned by { tA I X e A ). THEOREM. If the semigroup algebra kA has the same quotient field as k[t], then D(K) is Morita equivalent to A1. The present proof is due to S. P. Smith. The first result is well known, see [2, Lemma 2]. LEMMA 1. If K is a k-algebra and C a multiplicatively closed subset of K consisting of regular elements, then any differential operator on K has a unique extension to Kc. Hence D(K) may be regarded as a subalgebra of D(Kc). In addition, D(K) = {x e D(Kc) Ix K c K}. We let A1 k[q, p] be the k-algebra generated by q and p subject to the relation pq qp = 1, and regard A1 as D(k[t]) where q acts as a multiplication by t and p as d/dt. If K is a subalgebra of k[t] with the same quotient field k(t), then by Lemma 1, we can view both D(K) and A1 as subalgebras of the quotient ring Q of A1. We set I = {x E Aljx k[t] c K) and S = {x e QlxI c I). Then S_ EndA1(I), see [1, p. 69].

Some rings of differential operators which are Morita equivalent to the Weyl algebra 𝐴₁