A singular primitive ring

An example of a primitive ring with nonzero singular ideal is constructed. An example, due to B. Osofsky, of a semiprimitive ring with nonzero singular ideal is shown to be nonprimitive. All rings are associative with a unit element. All modules are unitary. R is a (right) primitive ring if it has a faithful irreducible right module. A right ideal of R is essential if it has nontrivial intersection with every nonzero right ideal of R. The singular ideal Z(R) is the set of elements of R which annihilate essential right ideals on the left. Equivalently Z(R) = lx E R: Vy (f 0) C R, 3z E R such that yz g 0, xyz = 0}. The existence of a primitive ring with nonzero singular ideal has been an open problem for several years. In [3] Osofsky constructed an example of a semiprimitive ring with nonzero singular ideal. Both Osofsky and Faith [1, p. 1281 conjectured tne existence of primitive rings with singular ideal. In this paper we construct such a ring, and also show that Osofsky's ring is not primitive. In proving the primitivity of the ring, we use several ideas from [2]. In particular we use Theorem 1. A ring is (right) primitive if and only if it has a proper right ideal M comaximal with every nonzero two-sided ideal of R; i.e. if J (; 0) is a two-sided ideal, then J + MR. The ring which we will construct is very similar to Osofsky's example in [31. Let F = Z2[X, Y.I, = 1, 2, ***, be the algebra over Z2 in noncommuting variables. An arbitrary monomial in F can be written as Presented to the Society, August 22, 1973 under the title A primitive ring with nonzero singular ideal; received by the editors July 31, 1973. AMS (MOS) subject classifications (1970). Primary 16A20; Secondary 16A08.