The Unitability of l-Prime Lattice-Ordered Rings with Squares Positive

Abstract

It is shown that an i-prime lattice-ordered ring with squares positive and an f-superunit can be embedded in a unital i-prime lattice-ordered ring with squares positive. In [1, p. 325] Steinberg asked if an i-prime i-ring with squares positive and an f-superunit can be embedded in a unital i-prime i-ring with squares positive. In this paper we show that the answer is yes. If R is a lattice-ordered ring (i-ring) and a E R+, then a is called an felement of R if b A c = 0 implies ab A c = ba A c = 0. Let T = T(R) = {a E R: lal is an f-element of R}. Then T is a convex f-subring of R, and R is a subdirect product of totally ordered T-T bimodules [2, Lemma 1]. R is an f-ring precisely when T = R. Throughout this paper T will denote the subring of f-elements of R. An element e > 0 of an i-ring R is called a superunit if ex > x and xe > x for each x E R+; e is an f-superunit if it is a superunit and an f-element. R is infinitesimal if x2 0 for each a in R. Let A be any ring and a E A,. If a satisfies ab = ba = nb for some fixed integer n and all b E A, then a is said to be an n-fier of A and n is said to have an n-fier a in A. Let K = {n E Z: n has n-fiers in A}. Then K is an ideal in the ring Z of integers. The ideal K is called the modal ideal of A; its nonnegative generator k is called the mode of A. If R is an f-ring with mode k > 0, then R has a unique k-fier x > 0 [3, III, Lemma 2.1]. Let R be an /-ring, and let S = S(R) = {a E R: lal > d, Vd E T+} and T' = {a E R: lal A d = 0, Vd E T+}. It is clear that T' is a convex /-subgroup of R. The referee has pointed out to us that the following lemma, on which our arguments are based, is a special case of [4, Lemma 6.2], and P. Conrad attributes it to A. H. Clifford. Received by the editors July 20, 1992 and, in revised form, October 30, 1992. 1991 Mathematics Subject Classification. Primary 06F25; Secondary 1 6A86.